From 53ede8c9bad73793c9431a51ac511eefa8c09eb2 Mon Sep 17 00:00:00 2001
From: Matthias Koeppe <mkoeppe@math.ucdavis.edu>
Date: Mon, 5 Dec 2022 14:21:20 -0800
Subject: [PATCH 01/10] sage.rings.finite_rings: Deprecate is_FiniteField,
 is_PrimeFiniteField

---
 src/sage/crypto/boolean_function.pyx             | 11 +++++++----
 src/sage/crypto/lfsr.py                          |  4 ++--
 .../dynamics/arithmetic_dynamics/affine_ds.py    | 10 +++++-----
 .../dynamics/arithmetic_dynamics/generic_ds.py   |  8 ++++----
 .../arithmetic_dynamics/projective_ds.py         | 14 +++++++-------
 src/sage/groups/cubic_braid.py                   |  4 ++--
 src/sage/groups/matrix_gps/orthogonal.py         |  8 ++++----
 src/sage/groups/matrix_gps/symplectic.py         |  4 ++--
 src/sage/groups/matrix_gps/unitary.py            |  8 ++++----
 src/sage/matrix/matrix_space.py                  |  4 ++--
 src/sage/modules/free_module.py                  |  2 +-
 src/sage/rings/algebraic_closure_finite_field.py |  6 +++---
 src/sage/rings/finite_rings/element_base.pyx     |  4 ++--
 .../rings/finite_rings/finite_field_base.pyx     | 11 ++++++++---
 .../finite_rings/finite_field_constructor.py     | 16 +++++++++++-----
 src/sage/rings/finite_rings/hom_finite_field.pyx |  8 ++++----
 .../finite_rings/hom_prime_finite_field.pyx      |  4 ++--
 src/sage/rings/finite_rings/homset.py            |  4 ++--
 src/sage/rings/morphism.pyx                      |  6 +++---
 src/sage/rings/polynomial/polynomial_element.pyx |  8 ++++----
 .../polynomial/polynomial_ring_constructor.py    |  4 ++--
 .../polynomial/polynomial_singular_interface.py  | 14 +++++++-------
 src/sage/schemes/affine/affine_homset.py         |  4 ++--
 src/sage/schemes/affine/affine_morphism.py       |  8 ++++----
 src/sage/schemes/affine/affine_space.py          | 16 ++++++++--------
 src/sage/schemes/curves/constructor.py           | 10 +++++-----
 src/sage/schemes/cyclic_covers/constructor.py    |  4 ++--
 src/sage/schemes/elliptic_curves/constructor.py  |  4 ++--
 src/sage/schemes/generic/algebraic_scheme.py     |  4 ++--
 .../schemes/hyperelliptic_curves/constructor.py  |  2 +-
 src/sage/schemes/plane_conics/constructor.py     |  4 ++--
 src/sage/schemes/product_projective/homset.py    |  4 ++--
 src/sage/schemes/product_projective/space.py     |  8 ++++----
 src/sage/schemes/projective/projective_homset.py |  4 ++--
 .../schemes/projective/projective_morphism.py    | 10 +++++-----
 src/sage/schemes/projective/projective_space.py  |  6 +++---
 .../schemes/projective/projective_subscheme.py   |  4 ++--
 src/sage/symbolic/ring.pyx                       |  4 ++--
 38 files changed, 136 insertions(+), 122 deletions(-)

diff --git a/src/sage/crypto/boolean_function.pyx b/src/sage/crypto/boolean_function.pyx
index a9ea665475c..10693a17992 100644
--- a/src/sage/crypto/boolean_function.pyx
+++ b/src/sage/crypto/boolean_function.pyx
@@ -38,8 +38,7 @@ from sage.rings.integer_ring import ZZ
 from sage.rings.integer cimport Integer
 from sage.rings.finite_rings.finite_field_constructor import GF
 from sage.rings.polynomial.pbori.pbori import BooleanPolynomial
-from sage.rings.finite_rings.finite_field_constructor import is_FiniteField
-from sage.rings.finite_rings.finite_field_givaro import FiniteField_givaro
+from sage.rings.finite_rings.finite_field_base import FiniteField
 from sage.rings.polynomial.polynomial_element import is_Polynomial
 
 from sage.misc.superseded import deprecated_function_alias
@@ -329,11 +328,15 @@ cdef class BooleanFunction(SageObject):
 
         elif is_Polynomial(x):
             K = x.base_ring()
-            if is_FiniteField(K) and K.characteristic() == 2:
+            if isinstance(K, FiniteField) and K.characteristic() == 2:
                 self._nvariables = K.degree()
                 bitset_init(self._truth_table, <mp_bitcnt_t> (1<<self._nvariables))
                 bitset_zero(self._truth_table)
-                if isinstance(K,FiniteField_givaro): #the ordering is not the same in this case
+                try:
+                    from sage.rings.finite_rings.finite_field_givaro import FiniteField_givaro
+                except ImportError:
+                    FiniteField_givaro = ()
+                if isinstance(K, FiniteField_givaro):  # the ordering is not the same in this case
                     for u in K:
                         bitset_set_to(self._truth_table, ZZ(u._vector_().list(),2) , (x(u)).trace())
                 else:
diff --git a/src/sage/crypto/lfsr.py b/src/sage/crypto/lfsr.py
index 40b469652ec..08553b62056 100644
--- a/src/sage/crypto/lfsr.py
+++ b/src/sage/crypto/lfsr.py
@@ -129,7 +129,7 @@
 
 from sage.structure.all import Sequence
 from sage.rings.all import Integer, PolynomialRing
-from sage.rings.finite_rings.finite_field_constructor import is_FiniteField
+from sage.rings.finite_rings.finite_field_base import FiniteField
 
 
 def lfsr_sequence(key, fill, n):
@@ -179,7 +179,7 @@ def lfsr_sequence(key, fill, n):
         raise TypeError("key must be a list")
     key = Sequence(key)
     F = key.universe()
-    if not is_FiniteField(F):
+    if not isinstance(F, FiniteField):
         raise TypeError("universe of sequence must be a finite field")
 
     s = fill
diff --git a/src/sage/dynamics/arithmetic_dynamics/affine_ds.py b/src/sage/dynamics/arithmetic_dynamics/affine_ds.py
index ccf1de6115d..e672314a9c0 100644
--- a/src/sage/dynamics/arithmetic_dynamics/affine_ds.py
+++ b/src/sage/dynamics/arithmetic_dynamics/affine_ds.py
@@ -43,8 +43,7 @@ class initialization directly.
 from sage.misc.cachefunc import cached_method
 from sage.misc.classcall_metaclass import typecall
 from sage.rings.integer import Integer
-from sage.rings.finite_rings.finite_field_constructor import is_PrimeFiniteField
-from sage.rings.finite_rings.finite_field_constructor import is_FiniteField
+from sage.rings.finite_rings.finite_field_base import FiniteField
 from sage.rings.fraction_field import FractionField
 from sage.rings.fraction_field import is_FractionField
 from sage.rings.quotient_ring import is_QuotientRing
@@ -251,7 +250,7 @@ def __classcall_private__(cls, morphism_or_polys, domain=None):
                 raise ValueError('domain and codomain do not agree')
             if R not in Fields():
                 return typecall(cls, polys, domain)
-            if is_FiniteField(R):
+            if isinstance(R, FiniteField):
                 return DynamicalSystem_affine_finite_field(polys, domain)
             return DynamicalSystem_affine_field(polys, domain)
         elif isinstance(morphism_or_polys,(list, tuple)):
@@ -299,7 +298,7 @@ def __classcall_private__(cls, morphism_or_polys, domain=None):
 
         if R not in Fields():
             return typecall(cls, polys, domain)
-        if is_FiniteField(R):
+        if isinstance(R, FiniteField):
             return DynamicalSystem_affine_finite_field(polys, domain)
         return DynamicalSystem_affine_field(polys, domain)
 
@@ -319,7 +318,8 @@ def __init__(self, polys_or_rat_fncts, domain):
         """
         L = polys_or_rat_fncts
         # Next attribute needed for _fast_eval and _fastpolys
-        self._is_prime_finite_field = is_PrimeFiniteField(L[0].base_ring())
+        R = L[0].base_ring()
+        self._is_prime_finite_field = isinstance(R, FiniteField) and R.is_prime_field()
         DynamicalSystem.__init__(self, L, domain)
 
     def __copy__(self):
diff --git a/src/sage/dynamics/arithmetic_dynamics/generic_ds.py b/src/sage/dynamics/arithmetic_dynamics/generic_ds.py
index 3e227942256..25d8645ae86 100644
--- a/src/sage/dynamics/arithmetic_dynamics/generic_ds.py
+++ b/src/sage/dynamics/arithmetic_dynamics/generic_ds.py
@@ -33,7 +33,7 @@ class initialization directly.
 from sage.schemes.affine.affine_space import is_AffineSpace
 from sage.schemes.affine.affine_subscheme import AlgebraicScheme_subscheme_affine
 from sage.rings.algebraic_closure_finite_field import AlgebraicClosureFiniteField_generic
-from sage.rings.finite_rings.finite_field_constructor import is_FiniteField
+from sage.rings.finite_rings.finite_field_base import FiniteField
 from sage.rings.qqbar import AlgebraicField_common
 from sage.schemes.berkovich.berkovich_space import is_Berkovich_Cp
 from sage.rings.rational_field import QQ
@@ -416,7 +416,7 @@ def field_of_definition_critical(self, return_embedding=False, simplify_all=Fals
             CR = CR.ring()
         x = CR.gen(0)
         poly = (g*CR(f).derivative(x) - f*CR(g).derivative(x)).univariate_polynomial()
-        if is_FiniteField(ds.base_ring()):
+        if isinstance(ds.base_ring(), FiniteField):
             return poly.splitting_field(names, map=return_embedding)
         else:
             K = poly.splitting_field(names, map=return_embedding, simplify_all=simplify_all)
@@ -536,7 +536,7 @@ def field_of_definition_periodic(self, n, formal=False, return_embedding=False,
             fn = ds.nth_iterate_map(n)
             f,g = fn[0].numerator(), fn[0].denominator()
             poly = (f - g*x).univariate_polynomial()
-        if is_FiniteField(ds.base_ring()):
+        if isinstance(ds.base_ring(), FiniteField):
             return poly.splitting_field(names, map=return_embedding)
         else:
             K = poly.splitting_field(names, map=return_embedding, simplify_all=simplify_all)
@@ -630,7 +630,7 @@ def field_of_definition_preimage(self, point, n, return_embedding=False, simplif
             #want the polynomial ring not the fraction field
             CR = CR.ring()
         poly = (f*point[1] - g*CR(point[0])).univariate_polynomial()
-        if is_FiniteField(ds.base_ring()):
+        if isinstance(ds.base_ring(), FiniteField):
             return poly.splitting_field(names, map=return_embedding)
         else:
             K = poly.splitting_field(names, map=return_embedding, simplify_all=simplify_all)
diff --git a/src/sage/dynamics/arithmetic_dynamics/projective_ds.py b/src/sage/dynamics/arithmetic_dynamics/projective_ds.py
index 79a4b06ebd6..30b11ce4261 100644
--- a/src/sage/dynamics/arithmetic_dynamics/projective_ds.py
+++ b/src/sage/dynamics/arithmetic_dynamics/projective_ds.py
@@ -73,8 +73,7 @@ class initialization directly.
 from sage.categories.finite_fields import FiniteFields
 from sage.rings.algebraic_closure_finite_field import AlgebraicClosureFiniteField_generic
 from sage.rings.complex_mpfr import ComplexField
-from sage.rings.finite_rings.finite_field_constructor import (is_FiniteField, GF,
-                                                              is_PrimeFiniteField)
+from sage.rings.finite_rings.finite_field_constructor import GF
 from sage.rings.finite_rings.integer_mod_ring import Zmod
 from sage.rings.fraction_field import (FractionField, is_FractionField, FractionField_1poly_field)
 from sage.rings.fraction_field_element import is_FractionFieldElement, FractionFieldElement
@@ -374,7 +373,7 @@ def __classcall_private__(cls, morphism_or_polys, domain=None, names=None):
                 raise ValueError('"domain" must be a projective scheme')
             if R not in Fields():
                 return typecall(cls, polys, domain)
-            if is_FiniteField(R):
+            if isinstance(R, FiniteField):
                 return DynamicalSystem_projective_finite_field(polys, domain)
             return DynamicalSystem_projective_field(polys, domain)
 
@@ -433,7 +432,7 @@ def __classcall_private__(cls, morphism_or_polys, domain=None, names=None):
                 if not all(split_d == domain._degree(f) for f in split_poly):
                     msg = 'polys (={}) must be multi-homogeneous of the same degrees (by component)'
                     raise TypeError(msg.format(polys))
-            if is_FiniteField(R):
+            if isinstance(R, FiniteField):
                 from sage.dynamics.arithmetic_dynamics.product_projective_ds import DynamicalSystem_product_projective_finite_field
                 return DynamicalSystem_product_projective_finite_field(polys, domain)
             return DynamicalSystem_product_projective(polys, domain)
@@ -451,7 +450,7 @@ def __classcall_private__(cls, morphism_or_polys, domain=None, names=None):
             raise ValueError('"domain" must be a projective scheme')
         if R not in Fields():
             return typecall(cls, polys, domain)
-        if is_FiniteField(R):
+        if isinstance(R, FiniteField):
                 return DynamicalSystem_projective_finite_field(polys, domain)
         return DynamicalSystem_projective_field(polys, domain)
 
@@ -470,8 +469,9 @@ def __init__(self, polys, domain):
                     (3/5*x^2 : y^2)
         """
         # Next attribute needed for _fast_eval and _fastpolys
-        self._is_prime_finite_field = is_PrimeFiniteField(polys[0].base_ring())
-        DynamicalSystem.__init__(self,polys,domain)
+        R = polys[0].base_ring()
+        self._is_prime_finite_field = isinstance(R, FiniteField) and R.is_prime_field()
+        DynamicalSystem.__init__(self, polys, domain)
 
     def __copy__(self):
         r"""
diff --git a/src/sage/groups/cubic_braid.py b/src/sage/groups/cubic_braid.py
index 4866cdc4f85..c8bfc659800 100644
--- a/src/sage/groups/cubic_braid.py
+++ b/src/sage/groups/cubic_braid.py
@@ -1543,10 +1543,10 @@ def as_matrix_group(self, root_bur=None, domain=None, characteristic=None, var='
                 self._classical_invariant_form = herm_form
 
         if unitary:
-            from sage.rings.finite_rings.finite_field_base import is_FiniteField
+            from sage.rings.finite_rings.finite_field_base import FiniteField
             from sage.groups.matrix_gps.unitary import GU
             d, d = herm_form.dimensions()
-            if is_FiniteField(domain):
+            if isinstance(domain, FiniteField):
                 base_group = GU(d, domain, var=domain.gen(), invariant_form=herm_form)
             else:
                 base_group = GU(d, domain, invariant_form=herm_form)
diff --git a/src/sage/groups/matrix_gps/orthogonal.py b/src/sage/groups/matrix_gps/orthogonal.py
index 8ff96051a8b..242c3f42529 100644
--- a/src/sage/groups/matrix_gps/orthogonal.py
+++ b/src/sage/groups/matrix_gps/orthogonal.py
@@ -85,7 +85,7 @@
 # ****************************************************************************
 
 from sage.rings.integer_ring import ZZ
-from sage.rings.finite_rings.finite_field_base import is_FiniteField
+from sage.rings.finite_rings.finite_field_base import FiniteField
 from sage.misc.latex import latex
 from sage.misc.cachefunc import cached_method
 from sage.groups.matrix_gps.named_group import (
@@ -127,7 +127,7 @@ def normalize_args_e(degree, ring, e):
         ...
         ValueError: must have e=-1 or e=1 for even degree
     """
-    if is_FiniteField(ring) and degree%2 == 0:
+    if isinstance(ring, FiniteField) and degree%2 == 0:
         if e not in (-1, +1):
             raise ValueError('must have e=-1 or e=1 for even degree')
     else:
@@ -171,7 +171,7 @@ def _OG(n, R, special, e=0, var='a', invariant_form=None):
     e = normalize_args_e(degree, ring, e)
 
     if invariant_form is not None:
-        if is_FiniteField(ring):
+        if isinstance(ring, FiniteField):
             raise NotImplementedError("invariant_form for finite groups is fixed by GAP")
 
     if e == 0:
@@ -202,7 +202,7 @@ def _OG(n, R, special, e=0, var='a', invariant_form=None):
                                                          latex(ring),
                                                          '+' if e == 1 else '-')
 
-    if is_FiniteField(ring):
+    if isinstance(ring, FiniteField):
         cmd  = '{0}O({1}, {2}, {3})'.format(ltx_prefix, e, degree, ring.order())
         return OrthogonalMatrixGroup_gap(degree, ring, False, name, ltx, cmd)
     else:
diff --git a/src/sage/groups/matrix_gps/symplectic.py b/src/sage/groups/matrix_gps/symplectic.py
index 3715f531913..8dd041af286 100644
--- a/src/sage/groups/matrix_gps/symplectic.py
+++ b/src/sage/groups/matrix_gps/symplectic.py
@@ -42,7 +42,7 @@
 
 from sage.misc.latex import latex
 from sage.misc.cachefunc import cached_method
-from sage.rings.finite_rings.finite_field_base import is_FiniteField
+from sage.rings.finite_rings.finite_field_base import FiniteField
 from sage.groups.matrix_gps.named_group import (
     normalize_args_vectorspace, normalize_args_invariant_form,
     NamedMatrixGroup_generic, NamedMatrixGroup_gap)
@@ -145,7 +145,7 @@ def Sp(n, R, var='a', invariant_form=None):
         raise ValueError('the degree must be even')
 
     if invariant_form is not None:
-        if is_FiniteField(ring):
+        if isinstance(ring, FiniteField):
             raise NotImplementedError("invariant_form for finite groups is fixed by GAP")
 
         invariant_form = normalize_args_invariant_form(ring, degree, invariant_form)
diff --git a/src/sage/groups/matrix_gps/unitary.py b/src/sage/groups/matrix_gps/unitary.py
index f4d8df525be..d8919f82695 100644
--- a/src/sage/groups/matrix_gps/unitary.py
+++ b/src/sage/groups/matrix_gps/unitary.py
@@ -50,7 +50,7 @@
 # ****************************************************************************
 
 from sage.rings.finite_rings.finite_field_constructor import GF
-from sage.rings.finite_rings.finite_field_base import is_FiniteField
+from sage.rings.finite_rings.finite_field_base import FiniteField
 from sage.misc.latex import latex
 from sage.misc.cachefunc import cached_method
 from sage.groups.matrix_gps.named_group import (
@@ -77,7 +77,7 @@ def finite_field_sqrt(ring):
         sage: finite_field_sqrt(GF(4, 'a'))
         2
     """
-    if not is_FiniteField(ring):
+    if not isinstance(ring, FiniteField):
         raise ValueError('not a finite field')
     q, rem = ring.cardinality().sqrtrem()
     if rem:
@@ -107,7 +107,7 @@ def _UG(n, R, special, var='a', invariant_form=None):
         latex_prefix ='S'
 
     degree, ring = normalize_args_vectorspace(n, R, var=var)
-    if is_FiniteField(ring):
+    if isinstance(ring, FiniteField):
         q = ring.cardinality()
         ring = GF(q**2, name=var)
         if invariant_form is not None:
@@ -134,7 +134,7 @@ def _UG(n, R, special, var='a', invariant_form=None):
         name = '{0} Unitary Group of degree {1} over {2}'.format(prefix, degree, ring)
         ltx  = r'\text{{{0}U}}_{{{1}}}({2})'.format(latex_prefix, degree, latex(ring))
 
-    if is_FiniteField(ring):
+    if isinstance(ring, FiniteField):
         cmd = '{0}U({1}, {2})'.format(latex_prefix, degree, q)
         return UnitaryMatrixGroup_gap(degree, ring, special, name, ltx, cmd)
     else:
diff --git a/src/sage/matrix/matrix_space.py b/src/sage/matrix/matrix_space.py
index cfb8ce85931..9d1639a7f06 100644
--- a/src/sage/matrix/matrix_space.py
+++ b/src/sage/matrix/matrix_space.py
@@ -43,7 +43,7 @@
 from sage.structure.parent import Parent
 from sage.structure.unique_representation import UniqueRepresentation
 import sage.rings.integer as integer
-import sage.rings.finite_rings.finite_field_constructor
+from sage.rings.finite_rings.finite_field_base import FiniteField
 import sage.misc.latex as latex
 import sage.modules.free_module
 
@@ -223,7 +223,7 @@ def get_matrix_class(R, nrows, ncols, sparse, implementation):
                     else:
                         return matrix_complex_double_dense.Matrix_complex_double_dense
 
-            elif sage.rings.finite_rings.finite_field_constructor.is_FiniteField(R):
+            elif isinstance(R, FiniteField):
                 if R.order() == 2:
                     try:
                         from . import matrix_mod2_dense
diff --git a/src/sage/modules/free_module.py b/src/sage/modules/free_module.py
index 7ad14bc1c8b..ff916cc4ad2 100644
--- a/src/sage/modules/free_module.py
+++ b/src/sage/modules/free_module.py
@@ -6348,7 +6348,7 @@ def _element_constructor_(self, e, *args, **kwds):
         """
         try:
             k = e.parent()
-            if finite_field.is_FiniteField(k) and k.base_ring() == self.base_ring() and k.degree() == self.degree():
+            if finite_field.isinstance(k, FiniteField) and k.base_ring() == self.base_ring() and k.degree() == self.degree():
                 return self(e._vector_())
         except AttributeError:
             pass
diff --git a/src/sage/rings/algebraic_closure_finite_field.py b/src/sage/rings/algebraic_closure_finite_field.py
index a0d5bac9d76..8577f1c613e 100644
--- a/src/sage/rings/algebraic_closure_finite_field.py
+++ b/src/sage/rings/algebraic_closure_finite_field.py
@@ -58,7 +58,7 @@
 from sage.misc.fast_methods import WithEqualityById
 
 from sage.rings.finite_rings.element_base import is_FiniteFieldElement
-from sage.rings.finite_rings.finite_field_base import is_FiniteField
+from sage.rings.finite_rings.finite_field_base import FiniteField
 from sage.rings.ring import Field
 from sage.structure.element import FieldElement
 from sage.structure.richcmp import richcmp
@@ -659,7 +659,7 @@ def _coerce_map_from_(self, other):
         """
         if other is self:
             return True
-        elif is_FiniteField(other) and self._subfield(other.degree()) is other:
+        elif isinstance(other, FiniteField) and self._subfield(other.degree()) is other:
             return True
         elif self._subfield(1).has_coerce_map_from(other):
             return True
@@ -1092,7 +1092,7 @@ def __init__(self, base_ring, name, category=None, lattice=None, use_database=Tr
             :meth:`AlgebraicClosureFiniteFieldElement.__init__`.
 
         """
-        if not (is_FiniteField(base_ring) and base_ring.is_prime_field()):
+        if not (isinstance(base_ring, FiniteField) and base_ring.is_prime_field()):
             raise NotImplementedError('algebraic closures of finite fields are only implemented for prime fields')
         from sage.rings.finite_rings.conway_polynomials import PseudoConwayLattice
         p = base_ring.characteristic()
diff --git a/src/sage/rings/finite_rings/element_base.pyx b/src/sage/rings/finite_rings/element_base.pyx
index 29ab05dd8ba..02ca50d18d1 100755
--- a/src/sage/rings/finite_rings/element_base.pyx
+++ b/src/sage/rings/finite_rings/element_base.pyx
@@ -37,8 +37,8 @@ def is_FiniteFieldElement(x):
         sage: is_FiniteFieldElement(GF(5)(2))
         True
     """
-    from sage.rings.finite_rings.finite_field_base import is_FiniteField
-    return isinstance(x, Element) and is_FiniteField(x.parent())
+    from sage.rings.finite_rings.finite_field_base import FiniteField
+    return isinstance(x, Element) and isinstance(x.parent(), FiniteField)
 
 
 cdef class FiniteRingElement(CommutativeRingElement):
diff --git a/src/sage/rings/finite_rings/finite_field_base.pyx b/src/sage/rings/finite_rings/finite_field_base.pyx
index 4c4f3cdb2e6..d13c93f23be 100644
--- a/src/sage/rings/finite_rings/finite_field_base.pyx
+++ b/src/sage/rings/finite_rings/finite_field_base.pyx
@@ -930,7 +930,7 @@ cdef class FiniteField(Field):
             sage: GF(3, 'a').is_prime_field()
             True
         """
-        return self.degree()==1
+        return self.degree() == 1
 
     def modulus(self):
         r"""
@@ -1356,7 +1356,7 @@ cdef class FiniteField(Field):
             return True
         if isinstance(R, sage.rings.abc.IntegerModRing) and self.characteristic().divides(R.characteristic()):
             return R.hom((self.one(),), check=False)
-        if is_FiniteField(R):
+        if isinstance(R, FiniteField):
             if R is self:
                 return True
             from .residue_field import ResidueField_generic
@@ -2115,10 +2115,13 @@ def is_FiniteField(R):
     Return whether the implementation of ``R`` has the interface provided by
     the standard finite field implementation.
 
+    This function is deprecated.
+
     EXAMPLES::
 
-        sage: from sage.rings.finite_rings.finite_field_base import is_FiniteField
+        sage: from sage.rings.finite_rings.finite_field_base import FiniteField
         sage: is_FiniteField(GF(9,'a'))
+        doctest:...: DeprecationWarning: the function is_FiniteField is deprecated; use isinstance(x, sage.rings.finite_rings.finite_field_base.FiniteField) instead
         True
         sage: is_FiniteField(GF(next_prime(10^10)))
         True
@@ -2128,4 +2131,6 @@ def is_FiniteField(R):
         sage: is_FiniteField(Integers(7))
         False
     """
+    from sage.misc.superseded import deprecation
+    deprecation(32664, "the function is_FiniteField is deprecated; use isinstance(x, sage.rings.finite_rings.finite_field_base.FiniteField) instead")
     return isinstance(R, FiniteField)
diff --git a/src/sage/rings/finite_rings/finite_field_constructor.py b/src/sage/rings/finite_rings/finite_field_constructor.py
index e42b2bed2dd..544e8125880 100644
--- a/src/sage/rings/finite_rings/finite_field_constructor.py
+++ b/src/sage/rings/finite_rings/finite_field_constructor.py
@@ -177,7 +177,7 @@
 from sage.rings.integer import Integer
 
 # the import below is just a redirection
-from sage.rings.finite_rings.finite_field_base import is_FiniteField
+from sage.rings.finite_rings.finite_field_base import FiniteField
 assert is_FiniteField  # just to silent pyflakes
 
 try:
@@ -791,24 +791,30 @@ def create_object(self, version, key, **kwds):
 
 def is_PrimeFiniteField(x):
     """
-    Returns True if x is a prime finite field.
+    Returns True if ``x`` is a prime finite field.
+
+    This function is deprecated.
 
     EXAMPLES::
 
-        sage: from sage.rings.finite_rings.finite_field_constructor import is_PrimeFiniteField
         sage: is_PrimeFiniteField(QQ)
+        doctest:...: DeprecationWarning: the function is_PrimeFiniteField is deprecated; use isinstance(x, sage.rings.finite_rings.finite_field_base.FiniteField) and x.is_prime_field() instead
         False
         sage: is_PrimeFiniteField(GF(7))
         True
-        sage: is_PrimeFiniteField(GF(7^2,'a'))
+        sage: is_PrimeFiniteField(GF(7^2, 'a'))
         False
-        sage: is_PrimeFiniteField(GF(next_prime(10^90,proof=False)))
+        sage: is_PrimeFiniteField(GF(next_prime(10^90, proof=False)))
         True
     """
+    from sage.misc.superseded import deprecation
+    deprecation(32664, "the function is_PrimeFiniteField is deprecated; use isinstance(x, sage.rings.finite_rings.finite_field_base.FiniteField) and x.is_prime_field() instead")
+
     from .finite_field_prime_modn import FiniteField_prime_modn
     from sage.rings.finite_rings.finite_field_base import FiniteField as FiniteField_generic
 
     return isinstance(x, FiniteField_prime_modn) or \
            (isinstance(x, FiniteField_generic) and x.degree() == 1)
 
+
 zech_log_bound = 2**16
diff --git a/src/sage/rings/finite_rings/hom_finite_field.pyx b/src/sage/rings/finite_rings/hom_finite_field.pyx
index 86d2401ceec..55f2cac6a1d 100644
--- a/src/sage/rings/finite_rings/hom_finite_field.pyx
+++ b/src/sage/rings/finite_rings/hom_finite_field.pyx
@@ -106,7 +106,7 @@ from sage.rings.integer cimport Integer
 from sage.categories.homset import Hom
 from sage.structure.element cimport Element
 
-from sage.rings.finite_rings.finite_field_base import is_FiniteField
+from sage.rings.finite_rings.finite_field_base import FiniteField
 from sage.rings.morphism cimport RingHomomorphism, RingHomomorphism_im_gens, FrobeniusEndomorphism_generic
 from sage.rings.finite_rings.finite_field_constructor import FiniteField
 
@@ -227,9 +227,9 @@ cdef class FiniteFieldHomomorphism_generic(RingHomomorphism_im_gens):
         """
         domain = parent.domain()
         codomain = parent.codomain()
-        if not is_FiniteField(domain):
+        if not isinstance(domain, FiniteField):
             raise TypeError("The domain is not a finite field or does not provide the required interface for finite fields")
-        if not is_FiniteField(codomain):
+        if not isinstance(codomain, FiniteField):
             raise TypeError("The codomain is not a finite field or does not provide the required interface for finite fields")
         if domain.characteristic() != codomain.characteristic() or codomain.degree() % domain.degree() != 0:
             raise ValueError("No embedding of %s into %s" % (domain, codomain))
@@ -522,7 +522,7 @@ cdef class FrobeniusEndomorphism_finite_field(FrobeniusEndomorphism_generic):
             ...
             TypeError: The domain is not a finite field or does not provide the required interface for finite fields
         """
-        if not is_FiniteField(domain):
+        if not isinstance(domain, FiniteField):
             raise TypeError("The domain is not a finite field or does not provide the required interface for finite fields")
         try:
             n = Integer(n)
diff --git a/src/sage/rings/finite_rings/hom_prime_finite_field.pyx b/src/sage/rings/finite_rings/hom_prime_finite_field.pyx
index b24c8bbfb3e..8120f5a4cd6 100644
--- a/src/sage/rings/finite_rings/hom_prime_finite_field.pyx
+++ b/src/sage/rings/finite_rings/hom_prime_finite_field.pyx
@@ -71,7 +71,7 @@ cdef class FiniteFieldHomomorphism_prime(FiniteFieldHomomorphism_generic):
             TypeError: The domain is not a finite prime field
         """
         domain = parent.domain()
-        if not is_FiniteField(domain) or not domain.is_prime_field():
+        if not isinstance(domain, FiniteField) or not domain.is_prime_field():
             raise TypeError("The domain is not a finite prime field")
         if section_class is None:
             section_class = SectionFiniteFieldHomomorphism_prime
@@ -100,7 +100,7 @@ cdef class FrobeniusEndomorphism_prime(FrobeniusEndomorphism_finite_field):
     fields (i.e. identity map :-).
     """
     def __init__(self, domain, power=1):
-        if not is_FiniteField(domain) or not domain.is_prime_field():
+        if not isinstance(domain, FiniteField) or not domain.is_prime_field():
             raise TypeError("The domain is not a finite prime field")
         FrobeniusEndomorphism_finite_field.__init__(self, Hom(domain, domain))
         self._order = 1
diff --git a/src/sage/rings/finite_rings/homset.py b/src/sage/rings/finite_rings/homset.py
index 0a7578a092e..ef629016983 100644
--- a/src/sage/rings/finite_rings/homset.py
+++ b/src/sage/rings/finite_rings/homset.py
@@ -37,7 +37,7 @@
 
 from sage.rings.homset import RingHomset_generic
 from sage.rings.finite_rings.hom_finite_field import FiniteFieldHomomorphism_generic
-from sage.rings.finite_rings.finite_field_base import is_FiniteField
+from sage.rings.finite_rings.finite_field_base import FiniteField
 from sage.rings.integer import Integer
 from sage.rings.morphism import RingHomomorphism_im_gens
 from sage.structure.sequence import Sequence
@@ -96,7 +96,7 @@ def __call__(self, im_gens, base_map=None, check=True):
             if self.domain().degree() == 1:
                 from sage.rings.finite_rings.hom_prime_finite_field import FiniteFieldHomomorphism_prime
                 return FiniteFieldHomomorphism_prime(self, im_gens, base_map=base_map, check=check)
-            if is_FiniteField(self.codomain()):
+            if isinstance(self.codomain(), FiniteField):
                 return FiniteFieldHomomorphism_generic(self, im_gens, base_map=base_map, check=check)
             # Currently, FiniteFieldHomomorphism_generic does not work if
             # the codomain is not derived from the finite field base class;
diff --git a/src/sage/rings/morphism.pyx b/src/sage/rings/morphism.pyx
index eb54dc4a900..00d810b5673 100644
--- a/src/sage/rings/morphism.pyx
+++ b/src/sage/rings/morphism.pyx
@@ -3160,7 +3160,7 @@ def _tensor_product_ring(B, A):
         ...
         ValueError: term ordering must be global
     """
-    from .finite_rings.finite_field_base import is_FiniteField
+    from .finite_rings.finite_field_base import FiniteField
     from .number_field.number_field_base import is_NumberField
     from .polynomial.multi_polynomial_ring import is_MPolynomialRing
     from .polynomial.polynomial_quotient_ring import is_PolynomialQuotientRing
@@ -3178,7 +3178,7 @@ def _tensor_product_ring(B, A):
     def term_order(A):
         # univariate rings do not have a term order
         if (is_PolynomialRing(A) or is_PolynomialQuotientRing(A)
-            or ((is_NumberField(A) or is_FiniteField(A))
+            or ((is_NumberField(A) or isinstance(A, FiniteField))
                 and not A.is_prime_field())):
             return TermOrder('lex', 1)
         try:
@@ -3201,7 +3201,7 @@ def _tensor_product_ring(B, A):
         elif is_QuotientRing(A):
             to_R = A.ambient().hom(R_gens_A, R, check=False)
             return list(to_R(A.defining_ideal()).gens())
-        elif ((is_NumberField(A) or is_FiniteField(A))
+        elif ((is_NumberField(A) or isinstance(A, FiniteField))
               and not A.is_prime_field()):
             to_R = A.polynomial_ring().hom(R_gens_A, R, check=False)
             return [to_R(A.polynomial())]
diff --git a/src/sage/rings/polynomial/polynomial_element.pyx b/src/sage/rings/polynomial/polynomial_element.pyx
index dc915077cf2..44d8282bd9c 100644
--- a/src/sage/rings/polynomial/polynomial_element.pyx
+++ b/src/sage/rings/polynomial/polynomial_element.pyx
@@ -4469,7 +4469,7 @@ cdef class Polynomial(CommutativeAlgebraElement):
         if not (ch == 0 or is_prime(ch)):
             raise NotImplementedError("factorization of polynomials over rings with composite characteristic is not implemented")
 
-        from sage.rings.finite_rings.finite_field_constructor import is_FiniteField
+        from sage.rings.finite_rings.finite_field_base import FiniteField
 
         n = None
 
@@ -4479,7 +4479,7 @@ cdef class Polynomial(CommutativeAlgebraElement):
             except PariError:
                 raise NotImplementedError
 
-        elif is_FiniteField(R):
+        elif isinstance(R, FiniteField):
             v = [x.__pari__("a") for x in self.list()]
             f = pari(v).Polrev()
             G = list(f.factor())
@@ -4747,7 +4747,7 @@ cdef class Polynomial(CommutativeAlgebraElement):
         name = normalize_names(1, names)[0]
 
         from sage.rings.number_field.number_field_base import is_NumberField
-        from sage.rings.finite_rings.finite_field_base import is_FiniteField
+        from sage.rings.finite_rings.finite_field_base import FiniteField
 
         f = self.monic()            # Given polynomial, made monic
         F = f.parent().base_ring()  # Base field
@@ -4758,7 +4758,7 @@ cdef class Polynomial(CommutativeAlgebraElement):
         if is_NumberField(F):
             from sage.rings.number_field.splitting_field import splitting_field
             return splitting_field(f, name, map, **kwds)
-        elif is_FiniteField(F):
+        elif isinstance(F, FiniteField):
             degree = lcm([f.degree() for f, _ in self.factor()])
             return F.extension(degree, name, map=map, **kwds)
 
diff --git a/src/sage/rings/polynomial/polynomial_ring_constructor.py b/src/sage/rings/polynomial/polynomial_ring_constructor.py
index e9a9365bda0..0a26b4cf0cf 100644
--- a/src/sage/rings/polynomial/polynomial_ring_constructor.py
+++ b/src/sage/rings/polynomial/polynomial_ring_constructor.py
@@ -26,7 +26,7 @@
 
 import sage.rings.abc
 from sage.rings.integer import Integer
-from sage.rings.finite_rings.finite_field_base import is_FiniteField
+from sage.rings.finite_rings.finite_field_base import FiniteField
 
 from sage.misc.cachefunc import weak_cached_function
 import sage.misc.weak_dict
@@ -701,7 +701,7 @@ def _single_variate(base_ring, name, sparse=None, implementation=None, order=Non
             specialized = polynomial_ring.PolynomialRing_dense_mod_p
         elif n > 1:  # Specialized code breaks for n == 1
             specialized = polynomial_ring.PolynomialRing_dense_mod_n
-    elif is_FiniteField(base_ring):
+    elif isinstance(base_ring, FiniteField):
         specialized = polynomial_ring.PolynomialRing_dense_finite_field
     elif isinstance(base_ring, padic_base_leaves.pAdicFieldCappedRelative):
         specialized = polynomial_ring.PolynomialRing_dense_padic_field_capped_relative
diff --git a/src/sage/rings/polynomial/polynomial_singular_interface.py b/src/sage/rings/polynomial/polynomial_singular_interface.py
index 40b77534d19..5d309ceb85c 100644
--- a/src/sage/rings/polynomial/polynomial_singular_interface.py
+++ b/src/sage/rings/polynomial/polynomial_singular_interface.py
@@ -44,7 +44,7 @@
 from sage.interfaces.singular import singular
 from sage.rings.rational_field import is_RationalField
 from sage.rings.function_field.function_field import RationalFunctionField
-from sage.rings.finite_rings.finite_field_base import is_FiniteField
+from sage.rings.finite_rings.finite_field_base import FiniteField
 from sage.rings.integer_ring import ZZ
 
 import sage.rings.finite_rings.finite_field_constructor
@@ -103,13 +103,13 @@ def _do_singular_init_(singular, base_ring, char, _vars, order):
 
     elif isinstance(base_ring, sage.rings.abc.IntegerModRing):
         char = base_ring.characteristic()
-        if sage.rings.finite_rings.finite_field_constructor.is_FiniteField(base_ring) and char <= 2147483647:
+        if isinstance(base_ring, FiniteField) and char <= 2147483647:
             return make_ring(str(char)), None
         if char.is_power_of(2):
             return make_ring(f"(integer,2,{char.nbits()-1})"), None
         return make_ring(f"(integer,{char})"), None
 
-    elif sage.rings.finite_rings.finite_field_constructor.is_FiniteField(base_ring):
+    elif isinstance(base_ring, FiniteField):
         # not the prime field!
         gen = str(base_ring.gen())
         R = make_ring(f"({char},{gen})")
@@ -148,7 +148,7 @@ def _do_singular_init_(singular, base_ring, char, _vars, order):
         if B.is_prime_field() or B is ZZ:
             return make_ring(f"({base_char},{gens})"), None
 
-        if is_FiniteField(B) and B.characteristic() <= 2147483647:
+        if isinstance(B, FiniteField) and B.characteristic() <= 2147483647:
             ext_gen = str(B.gen())
             _vars = '(' + ext_gen + ', ' + _vars[1:]
 
@@ -333,7 +333,7 @@ def _singular_(self, singular=singular):
             R._check_valid()
             if self.base_ring() is ZZ or self.base_ring().is_prime_field():
                 return R
-            if sage.rings.finite_rings.finite_field_constructor.is_FiniteField(self.base_ring()) or \
+            if isinstance(self.base_ring(), FiniteField) or \
                     (number_field.number_field_base.is_NumberField(self.base_ring()) and self.base_ring().is_absolute()):
                 R.set_ring()  # sorry for that, but needed for minpoly
                 if  singular.eval('minpoly') != f"({self.__minpoly})":
@@ -427,14 +427,14 @@ def can_convert_to_singular(R):
                                   sage.rings.abc.RealField, sage.rings.abc.ComplexField,
                                   sage.rings.abc.RealDoubleField, sage.rings.abc.ComplexDoubleField))):
         return True
-    elif sage.rings.finite_rings.finite_field_constructor.is_FiniteField(base_ring):
+    elif isinstance(base_ring, FiniteField):
         return base_ring.characteristic() <= 2147483647
     elif number_field.number_field_base.is_NumberField(base_ring):
         return base_ring.is_absolute()
     elif sage.rings.fraction_field.is_FractionField(base_ring):
         B = base_ring.base_ring()
         return (B.is_prime_field() or B is ZZ
-                or (is_FiniteField(B) and B.characteristic() <= 2147483647))
+                or (isinstance(B, FiniteField) and B.characteristic() <= 2147483647))
     elif isinstance(base_ring, RationalFunctionField):
         return base_ring.constant_field().is_prime_field()
     else:
diff --git a/src/sage/schemes/affine/affine_homset.py b/src/sage/schemes/affine/affine_homset.py
index 98b3b3a8994..914e54f7148 100644
--- a/src/sage/schemes/affine/affine_homset.py
+++ b/src/sage/schemes/affine/affine_homset.py
@@ -41,7 +41,7 @@
 from sage.rings.rational_field import is_RationalField
 from sage.categories.fields import Fields
 from sage.categories.number_fields import NumberFields
-from sage.rings.finite_rings.finite_field_constructor import is_FiniteField
+from sage.rings.finite_rings.finite_field_base import FiniteField
 from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
 from sage.schemes.generic.homset import SchemeHomset_points, SchemeHomset_generic
 
@@ -354,7 +354,7 @@ def points(self, **kwds):
                 raise TypeError("a positive bound B (= %s) must be specified"%B)
             from sage.schemes.affine.affine_rational_point import enum_affine_number_field
             return enum_affine_number_field(self, bound=B, tolerance=tol, precision=prec)
-        elif is_FiniteField(R):
+        elif isinstance(R, FiniteField):
             from sage.schemes.affine.affine_rational_point import enum_affine_finite_field
             return enum_affine_finite_field(self)
         else:
diff --git a/src/sage/schemes/affine/affine_morphism.py b/src/sage/schemes/affine/affine_morphism.py
index 1c4f2dff188..e6a8bad39bd 100644
--- a/src/sage/schemes/affine/affine_morphism.py
+++ b/src/sage/schemes/affine/affine_morphism.py
@@ -62,12 +62,11 @@
 from sage.arith.all import gcd
 
 from sage.rings.integer import Integer
-from sage.rings.finite_rings.finite_field_constructor import is_PrimeFiniteField
 from sage.rings.fraction_field import FractionField
 from sage.rings.fraction_field_element import FractionFieldElement
 from sage.rings.integer_ring import ZZ
 from sage.rings.rational_field import QQ
-from sage.rings.finite_rings.finite_field_constructor import is_FiniteField
+from sage.rings.finite_rings.finite_field_base import FiniteField
 
 from sage.schemes.generic.morphism import SchemeMorphism_polynomial
 
@@ -229,7 +228,8 @@ def __init__(self, parent, polys, check=True):
         SchemeMorphism_polynomial.__init__(self, parent, polys, check)
 
         # used in _fast_eval and _fastpolys
-        self._is_prime_finite_field = is_PrimeFiniteField(polys[0].base_ring())
+        R = polys[0].base_ring()
+        self._is_prime_finite_field = isinstance(R, FiniteField) and R.is_prime_field()
 
     def __call__(self, x, check=True):
         """
@@ -683,7 +683,7 @@ def as_dynamical_system(self):
         R = self.base_ring()
         if R not in _Fields:
             return DynamicalSystem_affine(list(self), self.domain())
-        if is_FiniteField(R):
+        if isinstance(R, FiniteField):
                 return DynamicalSystem_affine_finite_field(list(self), self.domain())
         return DynamicalSystem_affine_field(list(self), self.domain())
 
diff --git a/src/sage/schemes/affine/affine_space.py b/src/sage/schemes/affine/affine_space.py
index ba56ef9fc8d..52c13b29068 100644
--- a/src/sage/schemes/affine/affine_space.py
+++ b/src/sage/schemes/affine/affine_space.py
@@ -17,7 +17,7 @@
 from sage.rings.rational_field import is_RationalField
 from sage.rings.polynomial.polynomial_ring import is_PolynomialRing
 from sage.rings.polynomial.multi_polynomial_ring import is_MPolynomialRing
-from sage.rings.finite_rings.finite_field_constructor import is_FiniteField
+from sage.rings.finite_rings.finite_field_base import FiniteField
 from sage.categories.map import Map
 from sage.categories.fields import Fields
 from sage.categories.homset import Hom
@@ -132,7 +132,7 @@ def AffineSpace(n, R=None, names=None, ambient_projective_space=None,
         from sage.schemes.projective.projective_space import ProjectiveSpace
         ambient_projective_space = ProjectiveSpace(n, R)
     if R in _Fields:
-        if is_FiniteField(R):
+        if isinstance(R, FiniteField):
             return AffineSpace_finite_field(n, R, names,
                                             ambient_projective_space, default_embedding_index)
         else:
@@ -270,12 +270,12 @@ def rational_points(self, F=None):
             TypeError: second argument (= Integer Ring) must be a finite field
         """
         if F is None:
-            if not is_FiniteField(self.base_ring()):
-                raise TypeError("base ring (= %s) must be a finite field"%self.base_ring())
-            return [ P for P in self ]
-        elif not is_FiniteField(F):
-            raise TypeError("second argument (= %s) must be a finite field"%F)
-        return [ P for P in self.base_extend(F) ]
+            if not isinstance(self.base_ring(), FiniteField):
+                raise TypeError("base ring (= %s) must be a finite field" % self.base_ring())
+            return [P for P in self]
+        elif not isinstance(F, FiniteField):
+            raise TypeError("second argument (= %s) must be a finite field" % F)
+        return [P for P in self.base_extend(F)]
 
     def __eq__(self, right):
         """
diff --git a/src/sage/schemes/curves/constructor.py b/src/sage/schemes/curves/constructor.py
index 8a85dd9737c..9da795ac958 100644
--- a/src/sage/schemes/curves/constructor.py
+++ b/src/sage/schemes/curves/constructor.py
@@ -38,7 +38,7 @@
 
 from sage.rings.polynomial.multi_polynomial_element import is_MPolynomial
 from sage.rings.polynomial.multi_polynomial_ring import is_MPolynomialRing
-from sage.rings.finite_rings.finite_field_constructor import is_FiniteField
+from sage.rings.finite_rings.finite_field_base import FiniteField
 
 from sage.rings.rational_field import QQ
 
@@ -294,7 +294,7 @@ def Curve(F, A=None):
 
     if is_AffineSpace(A):
         if n != 2:
-            if is_FiniteField(k):
+            if isinstance(k, FiniteField):
                 if A.coordinate_ring().ideal(F).is_prime():
                     return IntegralAffineCurve_finite_field(A, F)
             if k in Fields():
@@ -307,7 +307,7 @@ def Curve(F, A=None):
             raise TypeError("need a single nonconstant polynomial to define a plane curve")
 
         F = F[0]
-        if is_FiniteField(k):
+        if isinstance(k, FiniteField):
             if _is_irreducible_and_reduced(F):
                 return IntegralAffinePlaneCurve_finite_field(A, F)
             return AffinePlaneCurve_finite_field(A, F)
@@ -321,7 +321,7 @@ def Curve(F, A=None):
         if n != 2:
             if not all(f.is_homogeneous() for f in F):
                 raise TypeError("polynomials defining a curve in a projective space must be homogeneous")
-            if is_FiniteField(k):
+            if isinstance(k, FiniteField):
                 if A.coordinate_ring().ideal(F).is_prime():
                     return IntegralProjectiveCurve_finite_field(A, F)
             if k in Fields():
@@ -339,7 +339,7 @@ def Curve(F, A=None):
         if not F.is_homogeneous():
             raise TypeError("{} is not a homogeneous polynomial".format(F))
 
-        if is_FiniteField(k):
+        if isinstance(k, FiniteField):
             if _is_irreducible_and_reduced(F):
                 return IntegralProjectivePlaneCurve_finite_field(A, F)
             return ProjectivePlaneCurve_finite_field(A, F)
diff --git a/src/sage/schemes/cyclic_covers/constructor.py b/src/sage/schemes/cyclic_covers/constructor.py
index 0966b0793d5..aba10277b19 100644
--- a/src/sage/schemes/cyclic_covers/constructor.py
+++ b/src/sage/schemes/cyclic_covers/constructor.py
@@ -9,7 +9,7 @@
 # *****************************************************************************
 
 from sage.rings.polynomial.polynomial_element import is_Polynomial
-from sage.rings.finite_rings.finite_field_constructor import is_FiniteField
+from sage.rings.finite_rings.finite_field_base import FiniteField
 from sage.schemes.affine.affine_space import AffineSpace
 from .cycliccover_generic import CyclicCover_generic
 from .cycliccover_finite_field import CyclicCover_finite_field
@@ -127,7 +127,7 @@ def CyclicCover(r, f, names=None, check_smooth=True):
         names = ["x", "y"]
     A2 = AffineSpace(2, R, names=names)
 
-    if is_FiniteField(R):
+    if isinstance(R, FiniteField):
         return CyclicCover_finite_field(A2, r, f, names=names)
     else:
         return CyclicCover_generic(A2, r, f, names=names)
diff --git a/src/sage/schemes/elliptic_curves/constructor.py b/src/sage/schemes/elliptic_curves/constructor.py
index 1ecdc082cd5..931779c8c7a 100644
--- a/src/sage/schemes/elliptic_curves/constructor.py
+++ b/src/sage/schemes/elliptic_curves/constructor.py
@@ -27,7 +27,7 @@
 
 import sage.rings.abc
 from sage.rings.polynomial.multi_polynomial_ring import is_MPolynomialRing
-from sage.rings.finite_rings.finite_field_constructor import is_FiniteField
+from sage.rings.finite_rings.finite_field_base import FiniteField
 from sage.rings.number_field.number_field import is_NumberField
 from sage.rings.polynomial.multi_polynomial_element import is_MPolynomial
 from sage.rings.ring import is_Ring
@@ -477,7 +477,7 @@ def create_object(self, version, key, **kwds):
         elif isinstance(R, sage.rings.abc.pAdicField):
             from .ell_padic_field import EllipticCurve_padic_field
             return EllipticCurve_padic_field(R, x)
-        elif is_FiniteField(R) or (isinstance(R, sage.rings.abc.IntegerModRing) and R.characteristic().is_prime()):
+        elif isinstance(R, FiniteField) or (isinstance(R, sage.rings.abc.IntegerModRing) and R.characteristic().is_prime()):
             from .ell_finite_field import EllipticCurve_finite_field
             return EllipticCurve_finite_field(R, x)
         elif R in _Fields:
diff --git a/src/sage/schemes/generic/algebraic_scheme.py b/src/sage/schemes/generic/algebraic_scheme.py
index 60d883905fe..aac33ed3042 100644
--- a/src/sage/schemes/generic/algebraic_scheme.py
+++ b/src/sage/schemes/generic/algebraic_scheme.py
@@ -132,7 +132,7 @@
 from sage.rings.integer_ring import ZZ
 from sage.rings.qqbar import QQbar
 from sage.rings.rational_field import is_RationalField
-from sage.rings.finite_rings.finite_field_constructor import is_FiniteField
+from sage.rings.finite_rings.finite_field_base import FiniteField
 from sage.rings.number_field.order import is_NumberFieldOrder
 
 from sage.misc.latex import latex
@@ -886,7 +886,7 @@ def rational_points(self, **kwds):
         if bound == 0:
             if is_RationalField(F):
                 raise TypeError("A positive bound (= %s) must be specified."%bound)
-            if not is_FiniteField(F):
+            if not isinstance(F, FiniteField):
                 raise TypeError("Argument F (= %s) must be a finite field."%F)
         pts = []
         for P in self.ambient_space().rational_points(F):
diff --git a/src/sage/schemes/hyperelliptic_curves/constructor.py b/src/sage/schemes/hyperelliptic_curves/constructor.py
index 54556e08755..590677e21a9 100644
--- a/src/sage/schemes/hyperelliptic_curves/constructor.py
+++ b/src/sage/schemes/hyperelliptic_curves/constructor.py
@@ -24,7 +24,7 @@
 
 import sage.rings.abc
 from sage.rings.rational_field import is_RationalField
-from sage.rings.finite_rings.finite_field_constructor import is_FiniteField
+from sage.rings.finite_rings.finite_field_base import FiniteField
 from sage.rings.polynomial.polynomial_element import is_Polynomial
 
 from sage.structure.dynamic_class import dynamic_class
diff --git a/src/sage/schemes/plane_conics/constructor.py b/src/sage/schemes/plane_conics/constructor.py
index a2a14190087..a962a392a63 100644
--- a/src/sage/schemes/plane_conics/constructor.py
+++ b/src/sage/schemes/plane_conics/constructor.py
@@ -31,7 +31,7 @@
 
 from sage.rings.ring import IntegralDomain
 from sage.rings.rational_field import is_RationalField
-from sage.rings.finite_rings.finite_field_constructor import is_FiniteField
+from sage.rings.finite_rings.finite_field_base import FiniteField
 from sage.rings.polynomial.multi_polynomial_element import is_MPolynomial
 from sage.rings.polynomial.polynomial_ring import is_PolynomialRing
 from sage.rings.polynomial.multi_polynomial_ring import is_MPolynomialRing
@@ -233,7 +233,7 @@ def Conic(base_field, F=None, names=None, unique=True):
 
     if F.parent().ngens() == 3:
         P2 = ProjectiveSpace(2, base_field, names)
-        if is_FiniteField(base_field):
+        if isinstance(base_field, FiniteField):
             return ProjectiveConic_finite_field(P2, F)
         if is_RationalField(base_field):
             return ProjectiveConic_rational_field(P2, F)
diff --git a/src/sage/schemes/product_projective/homset.py b/src/sage/schemes/product_projective/homset.py
index 7f4e1de0773..2b0119b1250 100644
--- a/src/sage/schemes/product_projective/homset.py
+++ b/src/sage/schemes/product_projective/homset.py
@@ -22,7 +22,7 @@
 from sage.categories.number_fields import NumberFields
 from sage.misc.mrange import xmrange
 from sage.misc.misc_c import prod
-from sage.rings.finite_rings.finite_field_constructor import is_FiniteField
+from sage.rings.finite_rings.finite_field_base import FiniteField
 from sage.rings.rational_field import is_RationalField
 from sage.schemes.generic.algebraic_scheme import AlgebraicScheme_subscheme
 from sage.schemes.generic.homset import SchemeHomset_points
@@ -228,7 +228,7 @@ def points(self, **kwds):
                 raise TypeError("a positive bound B (= %s) must be specified"%B)
             from sage.schemes.product_projective.rational_point import enum_product_projective_number_field
             return enum_product_projective_number_field(self, bound=B)
-        elif is_FiniteField(R):
+        elif isinstance(R, FiniteField):
             from sage.schemes.product_projective.rational_point import enum_product_projective_finite_field
             return enum_product_projective_finite_field(self)
         else:
diff --git a/src/sage/schemes/product_projective/space.py b/src/sage/schemes/product_projective/space.py
index 79bd69f671e..ac94e72cd0c 100644
--- a/src/sage/schemes/product_projective/space.py
+++ b/src/sage/schemes/product_projective/space.py
@@ -45,7 +45,7 @@
 from sage.rings.integer import Integer
 from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
 from sage.rings.rational_field import QQ
-from sage.rings.finite_rings.finite_field_constructor import is_FiniteField
+from sage.rings.finite_rings.finite_field_base import FiniteField
 from sage.categories.fields import Fields
 from sage.categories.commutative_rings import CommutativeRings
 from sage.rings.polynomial.polydict import ETuple
@@ -137,7 +137,7 @@ def ProductProjectiveSpaces(n, R=None, names='x'):
                 raise AttributeError("components must be over the same base ring")
             N.append(PS.dimension_relative())
             names += PS.variable_names()
-        if is_FiniteField(R):
+        if isinstance(R, FiniteField):
             X = ProductProjectiveSpaces_finite_field(N, R, names)
         elif R in Fields():
             X = ProductProjectiveSpaces_field(N, R, names)
@@ -162,7 +162,7 @@ def ProductProjectiveSpaces(n, R=None, names='x'):
             else:
                 n_vars = sum(1+d for d in n)
                 names = normalize_names(n_vars, name_list)
-        if is_FiniteField(R):
+        if isinstance(R, FiniteField):
             X = ProductProjectiveSpaces_finite_field(n, R, names)
         elif R in Fields():
             X = ProductProjectiveSpaces_field(n, R, names)
@@ -1302,6 +1302,6 @@ def rational_points(self, F=None):
         """
         if F is None:
             return list(self)
-        elif not is_FiniteField(F):
+        elif not isinstance(F, FiniteField):
             raise TypeError("second argument (= %s) must be a finite field"%F)
         return list(self.base_extend(F))
diff --git a/src/sage/schemes/projective/projective_homset.py b/src/sage/schemes/projective/projective_homset.py
index 3ffd4a6b6e4..3ee0b78876e 100644
--- a/src/sage/schemes/projective/projective_homset.py
+++ b/src/sage/schemes/projective/projective_homset.py
@@ -47,7 +47,7 @@
 from sage.rings.rational_field import is_RationalField
 from sage.categories.fields import Fields
 from sage.categories.number_fields import NumberFields
-from sage.rings.finite_rings.finite_field_constructor import is_FiniteField
+from sage.rings.finite_rings.finite_field_base import FiniteField
 from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
 from sage.schemes.generic.algebraic_scheme import AlgebraicScheme_subscheme
 from copy import copy
@@ -284,7 +284,7 @@ def points(self, **kwds):
                 raise TypeError("a positive bound B (= %s) must be specified"%B)
             from sage.schemes.projective.projective_rational_point import enum_projective_number_field
             return enum_projective_number_field(self, bound=B, tolerance=tol, precision=prec)
-        elif is_FiniteField(R):
+        elif isinstance(R, FiniteField):
             from sage.schemes.projective.projective_rational_point import enum_projective_finite_field
             return enum_projective_finite_field(self.extended_codomain())
         else:
diff --git a/src/sage/schemes/projective/projective_morphism.py b/src/sage/schemes/projective/projective_morphism.py
index 30afbb10ab1..45c1bb01b55 100644
--- a/src/sage/schemes/projective/projective_morphism.py
+++ b/src/sage/schemes/projective/projective_morphism.py
@@ -72,8 +72,7 @@
 import sage.rings.abc
 from sage.rings.integer import Integer
 from sage.rings.algebraic_closure_finite_field import AlgebraicClosureFiniteField_generic
-from sage.rings.finite_rings.finite_field_constructor import is_FiniteField
-from sage.rings.finite_rings.finite_field_constructor import is_PrimeFiniteField
+from sage.rings.finite_rings.finite_field_base import FiniteField
 from sage.rings.finite_rings.finite_field_constructor import GF
 from sage.rings.fraction_field import FractionField
 from sage.rings.integer_ring import ZZ
@@ -274,7 +273,8 @@ def __init__(self, parent, polys, check=True):
 
         SchemeMorphism_polynomial.__init__(self, parent, polys, check)
 
-        self._is_prime_finite_field = is_PrimeFiniteField(polys[0].base_ring())
+        R = polys[0].base_ring()
+        self._is_prime_finite_field = isinstance(R, FiniteField) and R.is_prime_field()
 
     def __call__(self, x, check=True):
         r"""
@@ -727,7 +727,7 @@ def as_dynamical_system(self):
         R = self.base_ring()
         if R not in _Fields:
             return DynamicalSystem_projective(list(self), self.domain())
-        if is_FiniteField(R):
+        if isinstance(R, FiniteField):
             return DynamicalSystem_projective_finite_field(list(self), self.domain())
         return DynamicalSystem_projective_field(list(self), self.domain())
 
@@ -1006,7 +1006,7 @@ def normalize_coordinates(self, **kwds):
         # scales by 1/gcd of the coefficients.
         if R in _NumberFields:
             O = R.maximal_order()
-        elif is_FiniteField(R):
+        elif isinstance(R, FiniteField):
             O = R
         elif isinstance(R, QuotientRing_generic):
             O = R.ring()
diff --git a/src/sage/schemes/projective/projective_space.py b/src/sage/schemes/projective/projective_space.py
index 670304af237..e215da942e1 100644
--- a/src/sage/schemes/projective/projective_space.py
+++ b/src/sage/schemes/projective/projective_space.py
@@ -81,7 +81,7 @@
 
 from sage.arith.misc import gcd, binomial
 
-from sage.rings.finite_rings.finite_field_constructor import is_FiniteField
+from sage.rings.finite_rings.finite_field_base import FiniteField
 from sage.rings.integer import Integer
 from sage.rings.integer_ring import ZZ
 from sage.rings.polynomial.multi_polynomial_ring import is_MPolynomialRing
@@ -256,7 +256,7 @@ def ProjectiveSpace(n, R=None, names=None):
     if R is None:
         R = ZZ  # default is the integers
     if R in _Fields:
-        if is_FiniteField(R):
+        if isinstance(R, FiniteField):
             return ProjectiveSpace_finite_field(n, R, names)
         if is_RationalField(R):
             return ProjectiveSpace_rational_field(n, R, names)
@@ -2253,7 +2253,7 @@ def rational_points(self, F=None):
         """
         if F is None:
             return [P for P in self]
-        elif not is_FiniteField(F):
+        elif not isinstance(F, FiniteField):
             raise TypeError("second argument (= %s) must be a finite field" % F)
         return [P for P in self.base_extend(F)]
 
diff --git a/src/sage/schemes/projective/projective_subscheme.py b/src/sage/schemes/projective/projective_subscheme.py
index 3c0faa498a6..2d7bdf2974a 100644
--- a/src/sage/schemes/projective/projective_subscheme.py
+++ b/src/sage/schemes/projective/projective_subscheme.py
@@ -29,7 +29,7 @@
 from sage.matrix.constructor import matrix
 
 from sage.rings.integer_ring import ZZ
-from sage.rings.finite_rings.finite_field_constructor import is_FiniteField
+from sage.rings.finite_rings.finite_field_base import FiniteField
 from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
 from sage.rings.rational_field import is_RationalField
 
@@ -982,7 +982,7 @@ def dual(self):
         from sage.libs.singular.function_factory import ff
 
         K = self.base_ring()
-        if not(is_RationalField(K) or is_FiniteField(K)):
+        if not(is_RationalField(K) or isinstance(K, FiniteField)):
             raise NotImplementedError("base ring must be QQ or a finite field")
         I = self.defining_ideal()
         m = I.ngens()
diff --git a/src/sage/symbolic/ring.pyx b/src/sage/symbolic/ring.pyx
index 2036a7331d4..9ecb93578d8 100644
--- a/src/sage/symbolic/ring.pyx
+++ b/src/sage/symbolic/ring.pyx
@@ -217,7 +217,7 @@ cdef class SymbolicRing(sage.rings.abc.SymbolicRing):
             from sage.rings.complex_mpfr import ComplexField
             from sage.rings.infinity import InfinityRing, UnsignedInfinityRing
             from sage.rings.real_lazy import RLF, CLF
-            from sage.rings.finite_rings.finite_field_base import is_FiniteField
+            from sage.rings.finite_rings.finite_field_base import FiniteField
 
             from sage.interfaces.maxima import Maxima
 
@@ -235,7 +235,7 @@ cdef class SymbolicRing(sage.rings.abc.SymbolicRing):
                                     sage.rings.abc.RealBallField,
                                     sage.rings.abc.ComplexBallField,
                                     sage.rings.abc.IntegerModRing))
-                  or is_FiniteField(R)):
+                  or isinstance(R, FiniteField)):
                 return True
             elif isinstance(R, GenericSymbolicSubring):
                 return True

From 8d4d8e58683173fc9b452f9f3a8d764785ba1b36 Mon Sep 17 00:00:00 2001
From: Matthias Koeppe <mkoeppe@math.ucdavis.edu>
Date: Mon, 5 Dec 2022 14:46:10 -0800
Subject: [PATCH 02/10] sage.rings.finite_rings: Make it a namespace package

---
 src/.relint.yml                         | 2 +-
 src/sage/rings/finite_rings/__init__.py | 0
 2 files changed, 1 insertion(+), 1 deletion(-)
 delete mode 100644 src/sage/rings/finite_rings/__init__.py

diff --git a/src/.relint.yml b/src/.relint.yml
index abadae0de28..d1534abfdd2 100644
--- a/src/.relint.yml
+++ b/src/.relint.yml
@@ -47,6 +47,6 @@
   hint: |
     Sage library code should not import from sage.PAC.KAGE.all when sage.PAC.KAGE is an implicit
     Hint: namespace package. Type import_statements("SOME_IDENTIFIER") to find a more specific import.
-  pattern: 'from\s+sage(|[.](arith|categories|combinat|ext|graphs(|[.]decompositions)|interfaces|libs|matrix|misc|numerical(|[.]backends)|rings|sets))[.]all\s+import'
+  pattern: 'from\s+sage(|[.](arith|categories|combinat|ext|graphs(|[.]decompositions)|interfaces|libs|matrix|misc|numerical(|[.]backends)|rings(|[.]finite_rings)|sets))[.]all\s+import'
   filePattern: '.*[.](py|pyx|pxi)$'
   error: false # Make this a warning instead of an error for now
diff --git a/src/sage/rings/finite_rings/__init__.py b/src/sage/rings/finite_rings/__init__.py
deleted file mode 100644
index e69de29bb2d..00000000000

From 6ce26bc9f01f71faeb4a144a67b8487671cc45f5 Mon Sep 17 00:00:00 2001
From: Matthias Koeppe <mkoeppe@math.ucdavis.edu>
Date: Mon, 5 Dec 2022 15:16:51 -0800
Subject: [PATCH 03/10] 
 src/sage/rings/finite_rings/finite_field_constructor.py: Restore re-export of
 is_FiniteField

---
 src/sage/rings/finite_rings/finite_field_constructor.py | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/sage/rings/finite_rings/finite_field_constructor.py b/src/sage/rings/finite_rings/finite_field_constructor.py
index 544e8125880..a264e86ecd5 100644
--- a/src/sage/rings/finite_rings/finite_field_constructor.py
+++ b/src/sage/rings/finite_rings/finite_field_constructor.py
@@ -177,7 +177,7 @@
 from sage.rings.integer import Integer
 
 # the import below is just a redirection
-from sage.rings.finite_rings.finite_field_base import FiniteField
+from sage.rings.finite_rings.finite_field_base import is_FiniteField
 assert is_FiniteField  # just to silent pyflakes
 
 try:

From 98a5484775ac1ec596baa9a3739499cf22cc61f0 Mon Sep 17 00:00:00 2001
From: Matthias Koeppe <mkoeppe@math.ucdavis.edu>
Date: Mon, 5 Dec 2022 15:28:53 -0800
Subject: [PATCH 04/10] src/sage/rings/finite_rings/hom_finite_field.pyx:
 Import ABC as FiniteField_base

---
 src/sage/rings/finite_rings/hom_finite_field.pyx | 8 ++++----
 1 file changed, 4 insertions(+), 4 deletions(-)

diff --git a/src/sage/rings/finite_rings/hom_finite_field.pyx b/src/sage/rings/finite_rings/hom_finite_field.pyx
index 55f2cac6a1d..44dc3450be8 100644
--- a/src/sage/rings/finite_rings/hom_finite_field.pyx
+++ b/src/sage/rings/finite_rings/hom_finite_field.pyx
@@ -106,7 +106,7 @@ from sage.rings.integer cimport Integer
 from sage.categories.homset import Hom
 from sage.structure.element cimport Element
 
-from sage.rings.finite_rings.finite_field_base import FiniteField
+from sage.rings.finite_rings.finite_field_base import FiniteField as FiniteField_base
 from sage.rings.morphism cimport RingHomomorphism, RingHomomorphism_im_gens, FrobeniusEndomorphism_generic
 from sage.rings.finite_rings.finite_field_constructor import FiniteField
 
@@ -227,9 +227,9 @@ cdef class FiniteFieldHomomorphism_generic(RingHomomorphism_im_gens):
         """
         domain = parent.domain()
         codomain = parent.codomain()
-        if not isinstance(domain, FiniteField):
+        if not isinstance(domain, FiniteField_base):
             raise TypeError("The domain is not a finite field or does not provide the required interface for finite fields")
-        if not isinstance(codomain, FiniteField):
+        if not isinstance(codomain, FiniteField_base):
             raise TypeError("The codomain is not a finite field or does not provide the required interface for finite fields")
         if domain.characteristic() != codomain.characteristic() or codomain.degree() % domain.degree() != 0:
             raise ValueError("No embedding of %s into %s" % (domain, codomain))
@@ -522,7 +522,7 @@ cdef class FrobeniusEndomorphism_finite_field(FrobeniusEndomorphism_generic):
             ...
             TypeError: The domain is not a finite field or does not provide the required interface for finite fields
         """
-        if not isinstance(domain, FiniteField):
+        if not isinstance(domain, FiniteField_base):
             raise TypeError("The domain is not a finite field or does not provide the required interface for finite fields")
         try:
             n = Integer(n)

From 2e6b5efe57a925b6bd5c2d9b8ad49f7718e05af7 Mon Sep 17 00:00:00 2001
From: Matthias Koeppe <mkoeppe@math.ucdavis.edu>
Date: Mon, 5 Dec 2022 15:29:18 -0800
Subject: [PATCH 05/10] sage.rings.finite_rings: Deprecate is_FiniteFieldElent

---
 src/sage/crypto/mq/rijndael_gf.py                    | 12 +++---------
 src/sage/crypto/sbox.pyx                             |  5 ++---
 .../dynamics/arithmetic_dynamics/projective_ds.py    |  1 +
 src/sage/rings/algebraic_closure_finite_field.py     |  3 +--
 src/sage/rings/finite_rings/element_base.pyx         |  8 +++++++-
 src/sage/rings/padics/padic_template_element.pxi     |  5 +++--
 src/sage/schemes/elliptic_curves/ell_finite_field.py |  5 +++--
 7 files changed, 20 insertions(+), 19 deletions(-)

diff --git a/src/sage/crypto/mq/rijndael_gf.py b/src/sage/crypto/mq/rijndael_gf.py
index 154c84d146c..1e08fb8e998 100644
--- a/src/sage/crypto/mq/rijndael_gf.py
+++ b/src/sage/crypto/mq/rijndael_gf.py
@@ -424,7 +424,7 @@
 
 from sage.matrix.constructor import matrix
 from sage.matrix.constructor import column_matrix
-from sage.structure.element import Matrix
+from sage.structure.element import FieldElement, Matrix
 from sage.rings.finite_rings.finite_field_constructor import FiniteField
 from sage.structure.sage_object import SageObject
 from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
@@ -757,10 +757,7 @@ def _GF_to_hex(self, GF):
             sage: rgf._GF_to_hex(output)
             'e142cd5fcd9d6d94a3340793034391b5'
         """
-        from sage.rings.finite_rings.element_base import is_FiniteFieldElement
-        if not isinstance(GF, Matrix) and \
-           not isinstance(GF, list) and \
-           not is_FiniteFieldElement(GF):
+        if not isinstance(GF, (Matrix, list, FieldElement)):
             msg = ("keyword 'GF' must be a matrix over {0}, a list of "
                    "elements from {0}, or a single element from {0}")
             raise TypeError(msg.format(self._F))
@@ -883,10 +880,7 @@ def _GF_to_bin(self, GF):
             sage: rgf._GF_to_bin(output)
             '11011000000111111111100000011011110110000001111111111000000110111101100000011111111110000001101111011000000111111111100000011011'
         """
-        from sage.rings.finite_rings.element_base import is_FiniteFieldElement
-        if not isinstance(GF, Matrix) and \
-           not isinstance(GF, list) and \
-           not is_FiniteFieldElement(GF):
+        if not isinstance(GF, (Matrix, list, FieldElement)):
             msg = ("keyword 'GF' must be a matrix over {0}, a list of "
                    "elements from {0}, or a single element from {0}")
             raise TypeError(msg.format(self))
diff --git a/src/sage/crypto/sbox.pyx b/src/sage/crypto/sbox.pyx
index 77b7a904cc2..fb9a6f39da3 100644
--- a/src/sage/crypto/sbox.pyx
+++ b/src/sage/crypto/sbox.pyx
@@ -5,7 +5,7 @@ cimport cython
 from cysignals.memory cimport check_allocarray, sig_free
 
 from sage.structure.sage_object cimport SageObject
-from sage.structure.element cimport Element
+from sage.structure.element cimport Element, FieldElement
 
 from sage.combinat.integer_vector import IntegerVectors
 from sage.crypto.boolean_function import BooleanFunction
@@ -17,7 +17,6 @@ from sage.misc.functional import is_even
 from sage.misc.misc_c import prod as mul
 from sage.misc.superseded import deprecated_function_alias
 from sage.modules.free_module_element import vector
-from sage.rings.finite_rings.element_base import is_FiniteFieldElement
 from sage.rings.finite_rings.finite_field_constructor import FiniteField as GF
 from sage.rings.ideal import FieldIdeal, Ideal
 from sage.rings.integer_ring import ZZ
@@ -195,7 +194,7 @@ cdef class SBox(SageObject):
 
         _S_list = []
         for e in S:
-            if is_FiniteFieldElement(e):
+            if isinstance(e, FieldElement) and e.parent().is_finite():
                 e = e.polynomial().change_ring(ZZ).subs(e.parent().characteristic())
             _S_list.append(e)
         S = _S_list
diff --git a/src/sage/dynamics/arithmetic_dynamics/projective_ds.py b/src/sage/dynamics/arithmetic_dynamics/projective_ds.py
index 30b11ce4261..eb18d2fd0d3 100644
--- a/src/sage/dynamics/arithmetic_dynamics/projective_ds.py
+++ b/src/sage/dynamics/arithmetic_dynamics/projective_ds.py
@@ -73,6 +73,7 @@ class initialization directly.
 from sage.categories.finite_fields import FiniteFields
 from sage.rings.algebraic_closure_finite_field import AlgebraicClosureFiniteField_generic
 from sage.rings.complex_mpfr import ComplexField
+from sage.rings.finite_rings.finite_field_base import FiniteField
 from sage.rings.finite_rings.finite_field_constructor import GF
 from sage.rings.finite_rings.integer_mod_ring import Zmod
 from sage.rings.fraction_field import (FractionField, is_FractionField, FractionField_1poly_field)
diff --git a/src/sage/rings/algebraic_closure_finite_field.py b/src/sage/rings/algebraic_closure_finite_field.py
index 8577f1c613e..5d05d77a9c7 100644
--- a/src/sage/rings/algebraic_closure_finite_field.py
+++ b/src/sage/rings/algebraic_closure_finite_field.py
@@ -57,7 +57,6 @@
 from sage.misc.abstract_method import abstract_method
 from sage.misc.fast_methods import WithEqualityById
 
-from sage.rings.finite_rings.element_base import is_FiniteFieldElement
 from sage.rings.finite_rings.finite_field_base import FiniteField
 from sage.rings.ring import Field
 from sage.structure.element import FieldElement
@@ -91,7 +90,7 @@ def __init__(self, parent, value):
             and ``loads(dumps(x))``.
 
         """
-        if is_FiniteFieldElement(value):
+        if isinstance(value, FieldElement) and value.parent().is_finite():
             n = value.parent().degree()
         else:
             from sage.rings.integer import Integer
diff --git a/src/sage/rings/finite_rings/element_base.pyx b/src/sage/rings/finite_rings/element_base.pyx
index 02ca50d18d1..d92fbb792d2 100755
--- a/src/sage/rings/finite_rings/element_base.pyx
+++ b/src/sage/rings/finite_rings/element_base.pyx
@@ -25,18 +25,24 @@ from sage.misc.superseded import deprecated_function_alias
 
 def is_FiniteFieldElement(x):
     """
-    Returns if x is a finite field element.
+    Return True if ``x`` is a finite field element.
+
+    This function is deprecated.
 
     EXAMPLES::
 
         sage: from sage.rings.finite_rings.element_base import is_FiniteFieldElement
         sage: is_FiniteFieldElement(1)
+        doctest:...: DeprecationWarning: the function is_FiniteFieldElement is deprecated; use isinstance(x, sage.structure.element.FieldElement) and x.parent().is_finite() instead
         False
         sage: is_FiniteFieldElement(IntegerRing())
         False
         sage: is_FiniteFieldElement(GF(5)(2))
         True
     """
+    from sage.misc.superseded import deprecation
+    deprecation(32664, "the function is_FiniteFieldElement is deprecated; use isinstance(x, sage.structure.element.FieldElement) and x.parent().is_finite() instead")
+
     from sage.rings.finite_rings.finite_field_base import FiniteField
     return isinstance(x, Element) and isinstance(x.parent(), FiniteField)
 
diff --git a/src/sage/rings/padics/padic_template_element.pxi b/src/sage/rings/padics/padic_template_element.pxi
index fed8ac89f81..2356fe9b2e6 100644
--- a/src/sage/rings/padics/padic_template_element.pxi
+++ b/src/sage/rings/padics/padic_template_element.pxi
@@ -30,13 +30,14 @@ import sage.rings.finite_rings.integer_mod
 from cypari2.types cimport *
 from cypari2.gen cimport Gen as pari_gen
 from sage.libs.pari.convert_gmp cimport INT_to_mpz
+from sage.rings.finite_rings.finite_field_base import FiniteField
 from sage.rings.padics.common_conversion cimport get_ordp, get_preccap
 from sage.rings.integer cimport Integer
 from sage.rings.infinity import infinity
 from sage.rings.rational import Rational
 from sage.rings.padics.precision_error import PrecisionError
 from sage.rings.padics.misc import trim_zeros
-from sage.structure.element import canonical_coercion
+from sage.structure.element import canonical_coercion, FieldElement
 import itertools
 
 cdef long maxordp = (1L << (sizeof(long) * 8 - 2)) - 1
@@ -145,7 +146,7 @@ cdef class pAdicTemplateElement(pAdicGenericElement):
         elif sage.rings.finite_rings.integer_mod.is_IntegerMod(x):
             if not Integer(self.prime_pow.prime).divides(x.parent().order()):
                 raise TypeError("p does not divide modulus %s"%x.parent().order())
-        elif sage.rings.finite_rings.element_base.is_FiniteFieldElement(x):
+        elif isinstance(x, FieldElement) and isinstance(x.parent(), FiniteField):
             k = self.parent().residue_field()
             if not k.has_coerce_map_from(x.parent()):
                 raise NotImplementedError("conversion from finite fields which do not embed into the residue field not implemented")
diff --git a/src/sage/schemes/elliptic_curves/ell_finite_field.py b/src/sage/schemes/elliptic_curves/ell_finite_field.py
index 6676a1964c2..bd7aaf44c09 100644
--- a/src/sage/schemes/elliptic_curves/ell_finite_field.py
+++ b/src/sage/schemes/elliptic_curves/ell_finite_field.py
@@ -29,7 +29,8 @@
 from .constructor import EllipticCurve
 from sage.schemes.hyperelliptic_curves.hyperelliptic_finite_field import HyperellipticCurve_finite_field
 from sage.rings.all import Integer, ZZ, PolynomialRing, GF, polygen
-from sage.rings.finite_rings.element_base import is_FiniteFieldElement
+from sage.rings.finite_rings.finite_field_base import FiniteField
+from sage.structure.element import FieldElement
 import sage.groups.generic as generic
 from . import ell_point
 from sage.arith.all import gcd, lcm, binomial
@@ -1529,7 +1530,7 @@ def is_j_supersingular(j, proof=True):
         sage: [p for p in prime_range(100) if is_j_supersingular(GF(p)(123456))]
         [2, 3, 59, 89]
     """
-    if not is_FiniteFieldElement(j):
+    if not (isinstance(j, FieldElement) and isinstance(j.parent(), FiniteField)):
         raise ValueError("%s must be an element of a finite field" % j)
 
     F = j.parent()

From 3be7c975fb00896aef21d721eda69d6d22759385 Mon Sep 17 00:00:00 2001
From: Matthias Koeppe <mkoeppe@math.ucdavis.edu>
Date: Mon, 5 Dec 2022 16:30:09 -0800
Subject: [PATCH 06/10] Fixups

---
 src/sage/modules/free_module.py                         | 4 ++--
 src/sage/rings/finite_rings/element_base.pyx            | 1 +
 src/sage/rings/finite_rings/finite_field_base.pyx       | 1 +
 src/sage/rings/finite_rings/finite_field_constructor.py | 2 ++
 src/sage/schemes/hyperelliptic_curves/constructor.py    | 6 +++++-
 5 files changed, 11 insertions(+), 3 deletions(-)

diff --git a/src/sage/modules/free_module.py b/src/sage/modules/free_module.py
index ff916cc4ad2..6b2f5b87b27 100644
--- a/src/sage/modules/free_module.py
+++ b/src/sage/modules/free_module.py
@@ -183,7 +183,6 @@
 import sage.misc.latex as latex
 
 from sage.modules.module import Module
-import sage.rings.finite_rings.finite_field_constructor as finite_field
 import sage.rings.ring as ring
 import sage.rings.abc
 import sage.rings.integer_ring
@@ -195,6 +194,7 @@
 from sage.categories.infinite_enumerated_sets import InfiniteEnumeratedSets
 from sage.misc.lazy_attribute import lazy_attribute
 from sage.misc.randstate import current_randstate
+from sage.rings.finite_rings.finite_field_base import FiniteField
 from sage.structure.factory import UniqueFactory
 from sage.structure.sequence import Sequence
 from sage.structure.richcmp import (richcmp_method, rich_to_bool, richcmp,
@@ -6348,7 +6348,7 @@ def _element_constructor_(self, e, *args, **kwds):
         """
         try:
             k = e.parent()
-            if finite_field.isinstance(k, FiniteField) and k.base_ring() == self.base_ring() and k.degree() == self.degree():
+            if isinstance(k, FiniteField) and k.base_ring() == self.base_ring() and k.degree() == self.degree():
                 return self(e._vector_())
         except AttributeError:
             pass
diff --git a/src/sage/rings/finite_rings/element_base.pyx b/src/sage/rings/finite_rings/element_base.pyx
index d92fbb792d2..17ca80e9028 100755
--- a/src/sage/rings/finite_rings/element_base.pyx
+++ b/src/sage/rings/finite_rings/element_base.pyx
@@ -34,6 +34,7 @@ def is_FiniteFieldElement(x):
         sage: from sage.rings.finite_rings.element_base import is_FiniteFieldElement
         sage: is_FiniteFieldElement(1)
         doctest:...: DeprecationWarning: the function is_FiniteFieldElement is deprecated; use isinstance(x, sage.structure.element.FieldElement) and x.parent().is_finite() instead
+        See https://trac.sagemath.org/32664 for details.
         False
         sage: is_FiniteFieldElement(IntegerRing())
         False
diff --git a/src/sage/rings/finite_rings/finite_field_base.pyx b/src/sage/rings/finite_rings/finite_field_base.pyx
index d13c93f23be..09959aeae49 100644
--- a/src/sage/rings/finite_rings/finite_field_base.pyx
+++ b/src/sage/rings/finite_rings/finite_field_base.pyx
@@ -2122,6 +2122,7 @@ def is_FiniteField(R):
         sage: from sage.rings.finite_rings.finite_field_base import FiniteField
         sage: is_FiniteField(GF(9,'a'))
         doctest:...: DeprecationWarning: the function is_FiniteField is deprecated; use isinstance(x, sage.rings.finite_rings.finite_field_base.FiniteField) instead
+        See https://trac.sagemath.org/32664 for details.
         True
         sage: is_FiniteField(GF(next_prime(10^10)))
         True
diff --git a/src/sage/rings/finite_rings/finite_field_constructor.py b/src/sage/rings/finite_rings/finite_field_constructor.py
index a264e86ecd5..23ed63d8b30 100644
--- a/src/sage/rings/finite_rings/finite_field_constructor.py
+++ b/src/sage/rings/finite_rings/finite_field_constructor.py
@@ -797,8 +797,10 @@ def is_PrimeFiniteField(x):
 
     EXAMPLES::
 
+        sage: from sage.rings.finite_rings.finite_field_constructor import is_PrimeFiniteField
         sage: is_PrimeFiniteField(QQ)
         doctest:...: DeprecationWarning: the function is_PrimeFiniteField is deprecated; use isinstance(x, sage.rings.finite_rings.finite_field_base.FiniteField) and x.is_prime_field() instead
+        See https://trac.sagemath.org/32664 for details.
         False
         sage: is_PrimeFiniteField(GF(7))
         True
diff --git a/src/sage/schemes/hyperelliptic_curves/constructor.py b/src/sage/schemes/hyperelliptic_curves/constructor.py
index 590677e21a9..21afb10ef3a 100644
--- a/src/sage/schemes/hyperelliptic_curves/constructor.py
+++ b/src/sage/schemes/hyperelliptic_curves/constructor.py
@@ -249,7 +249,11 @@ def HyperellipticCurve(f, h=0, names=None, PP=None, check_squarefree=True):
 
     genus_classes = {2: HyperellipticCurve_g2}
 
-    is_pAdicField = lambda x: isinstance(x, sage.rings.abc.pAdicField)
+    def is_FiniteField(x):
+        return isinstance(x, FiniteField)
+
+    def is_pAdicField(x):
+        return isinstance(x, sage.rings.abc.pAdicField)
 
     fields = [
         ("FiniteField", is_FiniteField, HyperellipticCurve_finite_field),

From 97a363e8cc60fc67718fad16d16ce583f24a7e0d Mon Sep 17 00:00:00 2001
From: Matthias Koeppe <mkoeppe@math.ucdavis.edu>
Date: Mon, 5 Dec 2022 17:58:06 -0800
Subject: [PATCH 07/10] Fixups

---
 src/sage/interfaces/gap.py                             | 2 +-
 src/sage/rings/finite_rings/finite_field_base.pyx      | 2 +-
 src/sage/rings/finite_rings/finite_field_ntl_gf2e.py   | 1 +
 src/sage/rings/finite_rings/hom_prime_finite_field.pyx | 2 +-
 src/sage/symbolic/ring.pyx                             | 4 ++--
 5 files changed, 6 insertions(+), 5 deletions(-)

diff --git a/src/sage/interfaces/gap.py b/src/sage/interfaces/gap.py
index c80a847d7e9..e105d5845ca 100644
--- a/src/sage/interfaces/gap.py
+++ b/src/sage/interfaces/gap.py
@@ -1743,7 +1743,7 @@ def intmod_gap_to_sage(x):
         sage: b.parent()
         Ring of integers modulo 65537
     """
-    from sage.rings.finite_rings.all import FiniteField
+    from sage.rings.finite_rings.finite_field_constructor import FiniteField
     from sage.rings.finite_rings.integer_mod import Mod
     from sage.rings.integer import Integer
     s = str(x)
diff --git a/src/sage/rings/finite_rings/finite_field_base.pyx b/src/sage/rings/finite_rings/finite_field_base.pyx
index 09959aeae49..470bbdf0502 100644
--- a/src/sage/rings/finite_rings/finite_field_base.pyx
+++ b/src/sage/rings/finite_rings/finite_field_base.pyx
@@ -2119,7 +2119,7 @@ def is_FiniteField(R):
 
     EXAMPLES::
 
-        sage: from sage.rings.finite_rings.finite_field_base import FiniteField
+        sage: from sage.rings.finite_rings.finite_field_base import is_FiniteField
         sage: is_FiniteField(GF(9,'a'))
         doctest:...: DeprecationWarning: the function is_FiniteField is deprecated; use isinstance(x, sage.rings.finite_rings.finite_field_base.FiniteField) instead
         See https://trac.sagemath.org/32664 for details.
diff --git a/src/sage/rings/finite_rings/finite_field_ntl_gf2e.py b/src/sage/rings/finite_rings/finite_field_ntl_gf2e.py
index 82004502674..47723af64bd 100644
--- a/src/sage/rings/finite_rings/finite_field_ntl_gf2e.py
+++ b/src/sage/rings/finite_rings/finite_field_ntl_gf2e.py
@@ -52,6 +52,7 @@ def late_import():
     import sage.rings.polynomial.polynomial_element
     is_Polynomial = sage.rings.polynomial.polynomial_element.is_Polynomial
 
+
 class FiniteField_ntl_gf2e(FiniteField):
     """
     Finite Field of characteristic 2 and order `2^n`.
diff --git a/src/sage/rings/finite_rings/hom_prime_finite_field.pyx b/src/sage/rings/finite_rings/hom_prime_finite_field.pyx
index 8120f5a4cd6..617595ea383 100644
--- a/src/sage/rings/finite_rings/hom_prime_finite_field.pyx
+++ b/src/sage/rings/finite_rings/hom_prime_finite_field.pyx
@@ -33,7 +33,7 @@ from .hom_finite_field cimport SectionFiniteFieldHomomorphism_generic
 from .hom_finite_field cimport FiniteFieldHomomorphism_generic
 from .hom_finite_field cimport FrobeniusEndomorphism_finite_field
 
-from sage.rings.finite_rings.finite_field_base import FiniteField, is_FiniteField
+from sage.rings.finite_rings.finite_field_base import FiniteField
 from sage.rings.morphism cimport RingHomomorphism_im_gens
 
 
diff --git a/src/sage/symbolic/ring.pyx b/src/sage/symbolic/ring.pyx
index 9ecb93578d8..ce033910621 100644
--- a/src/sage/symbolic/ring.pyx
+++ b/src/sage/symbolic/ring.pyx
@@ -234,8 +234,8 @@ cdef class SymbolicRing(sage.rings.abc.SymbolicRing):
                                     sage.rings.abc.ComplexIntervalField,
                                     sage.rings.abc.RealBallField,
                                     sage.rings.abc.ComplexBallField,
-                                    sage.rings.abc.IntegerModRing))
-                  or isinstance(R, FiniteField)):
+                                    sage.rings.abc.IntegerModRing,
+                                    FiniteField))):
                 return True
             elif isinstance(R, GenericSymbolicSubring):
                 return True

From 0c014033f33a907da6ba7c48224102e0ddd82af3 Mon Sep 17 00:00:00 2001
From: Matthias Koeppe <mkoeppe@math.ucdavis.edu>
Date: Mon, 5 Dec 2022 18:53:12 -0800
Subject: [PATCH 08/10] Fixup: Elements of finite fields are not instances of
 FieldElement

---
 src/sage/crypto/mq/rijndael_gf.py                    | 10 +++++++---
 src/sage/crypto/sbox.pyx                             |  5 +++--
 src/sage/rings/algebraic_closure_finite_field.py     |  4 ++--
 src/sage/rings/padics/padic_template_element.pxi     |  4 ++--
 src/sage/schemes/elliptic_curves/ell_finite_field.py |  4 ++--
 5 files changed, 16 insertions(+), 11 deletions(-)

diff --git a/src/sage/crypto/mq/rijndael_gf.py b/src/sage/crypto/mq/rijndael_gf.py
index 1e08fb8e998..2dccf6b03b4 100644
--- a/src/sage/crypto/mq/rijndael_gf.py
+++ b/src/sage/crypto/mq/rijndael_gf.py
@@ -424,7 +424,7 @@
 
 from sage.matrix.constructor import matrix
 from sage.matrix.constructor import column_matrix
-from sage.structure.element import FieldElement, Matrix
+from sage.structure.element import Element, Matrix
 from sage.rings.finite_rings.finite_field_constructor import FiniteField
 from sage.structure.sage_object import SageObject
 from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
@@ -757,7 +757,9 @@ def _GF_to_hex(self, GF):
             sage: rgf._GF_to_hex(output)
             'e142cd5fcd9d6d94a3340793034391b5'
         """
-        if not isinstance(GF, (Matrix, list, FieldElement)):
+        if not isinstance(GF, Matrix) and \
+           not isinstance(GF, list) and \
+           not (isinstance(GF, Element) and isinstance(GF.parent(), FiniteField)):
             msg = ("keyword 'GF' must be a matrix over {0}, a list of "
                    "elements from {0}, or a single element from {0}")
             raise TypeError(msg.format(self._F))
@@ -880,7 +882,9 @@ def _GF_to_bin(self, GF):
             sage: rgf._GF_to_bin(output)
             '11011000000111111111100000011011110110000001111111111000000110111101100000011111111110000001101111011000000111111111100000011011'
         """
-        if not isinstance(GF, (Matrix, list, FieldElement)):
+        if not isinstance(GF, Matrix) and \
+           not isinstance(GF, list) and \
+           not (isinstance(GF, Element) and isinstance(GF.parent(), FiniteField)):
             msg = ("keyword 'GF' must be a matrix over {0}, a list of "
                    "elements from {0}, or a single element from {0}")
             raise TypeError(msg.format(self))
diff --git a/src/sage/crypto/sbox.pyx b/src/sage/crypto/sbox.pyx
index fb9a6f39da3..87683d79517 100644
--- a/src/sage/crypto/sbox.pyx
+++ b/src/sage/crypto/sbox.pyx
@@ -5,7 +5,7 @@ cimport cython
 from cysignals.memory cimport check_allocarray, sig_free
 
 from sage.structure.sage_object cimport SageObject
-from sage.structure.element cimport Element, FieldElement
+from sage.structure.element cimport Element
 
 from sage.combinat.integer_vector import IntegerVectors
 from sage.crypto.boolean_function import BooleanFunction
@@ -17,6 +17,7 @@ from sage.misc.functional import is_even
 from sage.misc.misc_c import prod as mul
 from sage.misc.superseded import deprecated_function_alias
 from sage.modules.free_module_element import vector
+from sage.rings.finite_rings.finite_field_base import FiniteField
 from sage.rings.finite_rings.finite_field_constructor import FiniteField as GF
 from sage.rings.ideal import FieldIdeal, Ideal
 from sage.rings.integer_ring import ZZ
@@ -194,7 +195,7 @@ cdef class SBox(SageObject):
 
         _S_list = []
         for e in S:
-            if isinstance(e, FieldElement) and e.parent().is_finite():
+            if isinstance(e, Element) and isinstance(e.parent(), FiniteField):
                 e = e.polynomial().change_ring(ZZ).subs(e.parent().characteristic())
             _S_list.append(e)
         S = _S_list
diff --git a/src/sage/rings/algebraic_closure_finite_field.py b/src/sage/rings/algebraic_closure_finite_field.py
index 5d05d77a9c7..c527bcb6dc4 100644
--- a/src/sage/rings/algebraic_closure_finite_field.py
+++ b/src/sage/rings/algebraic_closure_finite_field.py
@@ -59,7 +59,7 @@
 
 from sage.rings.finite_rings.finite_field_base import FiniteField
 from sage.rings.ring import Field
-from sage.structure.element import FieldElement
+from sage.structure.element import Element, FieldElement
 from sage.structure.richcmp import richcmp
 
 
@@ -90,7 +90,7 @@ def __init__(self, parent, value):
             and ``loads(dumps(x))``.
 
         """
-        if isinstance(value, FieldElement) and value.parent().is_finite():
+        if isinstance(value, Element) and isinstance(value.parent(), FiniteField):
             n = value.parent().degree()
         else:
             from sage.rings.integer import Integer
diff --git a/src/sage/rings/padics/padic_template_element.pxi b/src/sage/rings/padics/padic_template_element.pxi
index 2356fe9b2e6..72ff7784eb1 100644
--- a/src/sage/rings/padics/padic_template_element.pxi
+++ b/src/sage/rings/padics/padic_template_element.pxi
@@ -37,7 +37,7 @@ from sage.rings.infinity import infinity
 from sage.rings.rational import Rational
 from sage.rings.padics.precision_error import PrecisionError
 from sage.rings.padics.misc import trim_zeros
-from sage.structure.element import canonical_coercion, FieldElement
+from sage.structure.element import canonical_coercion, Element
 import itertools
 
 cdef long maxordp = (1L << (sizeof(long) * 8 - 2)) - 1
@@ -146,7 +146,7 @@ cdef class pAdicTemplateElement(pAdicGenericElement):
         elif sage.rings.finite_rings.integer_mod.is_IntegerMod(x):
             if not Integer(self.prime_pow.prime).divides(x.parent().order()):
                 raise TypeError("p does not divide modulus %s"%x.parent().order())
-        elif isinstance(x, FieldElement) and isinstance(x.parent(), FiniteField):
+        elif isinstance(x, Element) and isinstance(x.parent(), FiniteField):
             k = self.parent().residue_field()
             if not k.has_coerce_map_from(x.parent()):
                 raise NotImplementedError("conversion from finite fields which do not embed into the residue field not implemented")
diff --git a/src/sage/schemes/elliptic_curves/ell_finite_field.py b/src/sage/schemes/elliptic_curves/ell_finite_field.py
index bd7aaf44c09..863c56b22ea 100644
--- a/src/sage/schemes/elliptic_curves/ell_finite_field.py
+++ b/src/sage/schemes/elliptic_curves/ell_finite_field.py
@@ -30,7 +30,7 @@
 from sage.schemes.hyperelliptic_curves.hyperelliptic_finite_field import HyperellipticCurve_finite_field
 from sage.rings.all import Integer, ZZ, PolynomialRing, GF, polygen
 from sage.rings.finite_rings.finite_field_base import FiniteField
-from sage.structure.element import FieldElement
+from sage.structure.element import Element
 import sage.groups.generic as generic
 from . import ell_point
 from sage.arith.all import gcd, lcm, binomial
@@ -1530,7 +1530,7 @@ def is_j_supersingular(j, proof=True):
         sage: [p for p in prime_range(100) if is_j_supersingular(GF(p)(123456))]
         [2, 3, 59, 89]
     """
-    if not (isinstance(j, FieldElement) and isinstance(j.parent(), FiniteField)):
+    if not (isinstance(j, Element) and isinstance(j.parent(), FiniteField)):
         raise ValueError("%s must be an element of a finite field" % j)
 
     F = j.parent()

From 9f4d1c4bd227c145d964bff1a4f40305f780be2d Mon Sep 17 00:00:00 2001
From: Matthias Koeppe <mkoeppe@math.ucdavis.edu>
Date: Mon, 5 Dec 2022 19:26:05 -0800
Subject: [PATCH 09/10] src/sage/crypto/mq/rijndael_gf.py: Fix imports

---
 src/sage/crypto/mq/rijndael_gf.py | 5 +++--
 1 file changed, 3 insertions(+), 2 deletions(-)

diff --git a/src/sage/crypto/mq/rijndael_gf.py b/src/sage/crypto/mq/rijndael_gf.py
index 2dccf6b03b4..6bcb86f4d21 100644
--- a/src/sage/crypto/mq/rijndael_gf.py
+++ b/src/sage/crypto/mq/rijndael_gf.py
@@ -425,6 +425,7 @@
 from sage.matrix.constructor import matrix
 from sage.matrix.constructor import column_matrix
 from sage.structure.element import Element, Matrix
+from sage.rings.finite_rings.finite_field_base import FiniteField as FiniteField_base
 from sage.rings.finite_rings.finite_field_constructor import FiniteField
 from sage.structure.sage_object import SageObject
 from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
@@ -759,7 +760,7 @@ def _GF_to_hex(self, GF):
         """
         if not isinstance(GF, Matrix) and \
            not isinstance(GF, list) and \
-           not (isinstance(GF, Element) and isinstance(GF.parent(), FiniteField)):
+           not (isinstance(GF, Element) and isinstance(GF.parent(), FiniteField_base)):
             msg = ("keyword 'GF' must be a matrix over {0}, a list of "
                    "elements from {0}, or a single element from {0}")
             raise TypeError(msg.format(self._F))
@@ -884,7 +885,7 @@ def _GF_to_bin(self, GF):
         """
         if not isinstance(GF, Matrix) and \
            not isinstance(GF, list) and \
-           not (isinstance(GF, Element) and isinstance(GF.parent(), FiniteField)):
+           not (isinstance(GF, Element) and isinstance(GF.parent(), FiniteField_base)):
             msg = ("keyword 'GF' must be a matrix over {0}, a list of "
                    "elements from {0}, or a single element from {0}")
             raise TypeError(msg.format(self))

From 40a7d574bce11dcf62575c276a0aa8ac76c55e56 Mon Sep 17 00:00:00 2001
From: Matthias Koeppe <mkoeppe@math.ucdavis.edu>
Date: Mon, 20 Feb 2023 18:37:07 -0800
Subject: [PATCH 10/10] src/sage/rings/finite_rings: Update doctest output
 (trac->github)

---
 src/sage/rings/finite_rings/element_base.pyx            | 2 +-
 src/sage/rings/finite_rings/finite_field_base.pyx       | 2 +-
 src/sage/rings/finite_rings/finite_field_constructor.py | 2 +-
 3 files changed, 3 insertions(+), 3 deletions(-)

diff --git a/src/sage/rings/finite_rings/element_base.pyx b/src/sage/rings/finite_rings/element_base.pyx
index 17ca80e9028..4f785ebf181 100755
--- a/src/sage/rings/finite_rings/element_base.pyx
+++ b/src/sage/rings/finite_rings/element_base.pyx
@@ -34,7 +34,7 @@ def is_FiniteFieldElement(x):
         sage: from sage.rings.finite_rings.element_base import is_FiniteFieldElement
         sage: is_FiniteFieldElement(1)
         doctest:...: DeprecationWarning: the function is_FiniteFieldElement is deprecated; use isinstance(x, sage.structure.element.FieldElement) and x.parent().is_finite() instead
-        See https://trac.sagemath.org/32664 for details.
+        See https://github.com/sagemath/sage/issues/32664 for details.
         False
         sage: is_FiniteFieldElement(IntegerRing())
         False
diff --git a/src/sage/rings/finite_rings/finite_field_base.pyx b/src/sage/rings/finite_rings/finite_field_base.pyx
index 1ec0f276289..1bdb3f48bb1 100644
--- a/src/sage/rings/finite_rings/finite_field_base.pyx
+++ b/src/sage/rings/finite_rings/finite_field_base.pyx
@@ -2122,7 +2122,7 @@ def is_FiniteField(R):
         sage: from sage.rings.finite_rings.finite_field_base import is_FiniteField
         sage: is_FiniteField(GF(9,'a'))
         doctest:...: DeprecationWarning: the function is_FiniteField is deprecated; use isinstance(x, sage.rings.finite_rings.finite_field_base.FiniteField) instead
-        See https://trac.sagemath.org/32664 for details.
+        See https://github.com/sagemath/sage/issues/32664 for details.
         True
         sage: is_FiniteField(GF(next_prime(10^10)))
         True
diff --git a/src/sage/rings/finite_rings/finite_field_constructor.py b/src/sage/rings/finite_rings/finite_field_constructor.py
index 23ed63d8b30..9a3f55ab489 100644
--- a/src/sage/rings/finite_rings/finite_field_constructor.py
+++ b/src/sage/rings/finite_rings/finite_field_constructor.py
@@ -800,7 +800,7 @@ def is_PrimeFiniteField(x):
         sage: from sage.rings.finite_rings.finite_field_constructor import is_PrimeFiniteField
         sage: is_PrimeFiniteField(QQ)
         doctest:...: DeprecationWarning: the function is_PrimeFiniteField is deprecated; use isinstance(x, sage.rings.finite_rings.finite_field_base.FiniteField) and x.is_prime_field() instead
-        See https://trac.sagemath.org/32664 for details.
+        See https://github.com/sagemath/sage/issues/32664 for details.
         False
         sage: is_PrimeFiniteField(GF(7))
         True