Total refactor of any_root() to solve issue in characteristic two and clean up the code

src/sage/rings/finite_rings/conway_polynomials.py

@@ -136,7 +136,7 @@ class PseudoConwayLattice(WithEqualityById, SageObject):
         sage: # needs sage.rings.finite_rings
         sage: from sage.rings.finite_rings.conway_polynomials import PseudoConwayLattice
         sage: PCL = PseudoConwayLattice(2, use_database=False)
-        sage: PCL.polynomial(3)
+        sage: PCL.polynomial(3)   # random
         x^3 + x + 1
 
     TESTS::
@@ -164,16 +164,16 @@ def __init__(self, p, use_database=True):
             sage: # needs sage.rings.finite_rings
             sage: from sage.rings.finite_rings.conway_polynomials import PseudoConwayLattice
             sage: PCL = PseudoConwayLattice(3)
-            sage: PCL.polynomial(3)
+            sage: PCL.polynomial(3)  # random
             x^3 + 2*x + 1
 
             sage: # needs sage.rings.finite_rings
             sage: PCL = PseudoConwayLattice(5, use_database=False)
-            sage: PCL.polynomial(12)
+            sage: PCL.polynomial(12)  # random
             x^12 + 4*x^11 + 2*x^10 + 4*x^9 + 2*x^8 + 2*x^7 + 4*x^6 + x^5 + 2*x^4 + 2*x^2 + x + 2
-            sage: PCL.polynomial(6)
+            sage: PCL.polynomial(6)   # random
             x^6 + x^5 + 4*x^4 + 3*x^3 + 3*x^2 + 2*x + 2
-            sage: PCL.polynomial(11)
+            sage: PCL.polynomial(11)  # random
             x^11 + x^6 + 3*x^3 + 4*x + 3
         """
         self.p = p
@@ -215,11 +215,11 @@ def polynomial(self, n):
             sage: # needs sage.rings.finite_rings
             sage: from sage.rings.finite_rings.conway_polynomials import PseudoConwayLattice
             sage: PCL = PseudoConwayLattice(2, use_database=False)
-            sage: PCL.polynomial(3)
+            sage: PCL.polynomial(3)   # random
             x^3 + x + 1
-            sage: PCL.polynomial(4)
+            sage: PCL.polynomial(4)   # random
             x^4 + x^3 + 1
-            sage: PCL.polynomial(60)
+            sage: PCL.polynomial(60)  # random
             x^60 + x^59 + x^58 + x^55 + x^54 + x^53 + x^52 + x^51 + x^48 + x^46 + x^45 + x^42 + x^41 + x^39 + x^38 + x^37 + x^35 + x^32 + x^31 + x^30 + x^28 + x^24 + x^22 + x^21 + x^18 + x^17 + x^16 + x^15 + x^14 + x^10 + x^8 + x^7 + x^5 + x^3 + x^2 + x + 1
         """
         if n in self.nodes:
@@ -239,7 +239,7 @@ def polynomial(self, n):
         # TODO: something like the following
         # gcds = [n.gcd(d) for d in self.nodes.keys()]
         # xi = { m: (...) for m in gcds }
-        xi = {q: self.polynomial(n//q).any_root(K, -n//q, assume_squarefree=True)
+        xi = {q: self.polynomial(n//q).any_root(K, n//q, assume_squarefree=True, assume_distinct_deg=True)
               for q in n.prime_divisors()}
 
         # The following is needed to ensure that in the concrete instantiation
@@ -402,9 +402,9 @@ def _frobenius_shift(K, generators, check_only=False):
         sage: f20 = x^20 + x^19 + x^15 + x^13 + x^12 + x^11 + x^9 + x^8 + x^7 + x^4 + x^2 + x + 1
         sage: f12 = x^12 + x^10 + x^9 + x^8 + x^4 + x^2 + 1
         sage: K.<a> = GF(2^60, modulus='first_lexicographic')
-        sage: x30 = f30.any_root(K)
-        sage: x20 = f20.any_root(K)
-        sage: x12 = f12.any_root(K)
+        sage: x30 = f30.roots(K, multiplicities=False)[0]
+        sage: x20 = f20.roots(K, multiplicities=False)[0]
+        sage: x12 = f12.roots(K, multiplicities=False)[0]
         sage: generators = {2: x30, 3: x20, 5: x12}
         sage: from sage.rings.finite_rings.conway_polynomials import _frobenius_shift, _find_pow_of_frobenius
         sage: _frobenius_shift(K, generators)

src/sage/rings/finite_rings/hom_finite_field.pyx

@@ -16,32 +16,35 @@ Construction of an embedding::
 
     sage: k.<t> = GF(3^7)
     sage: K.<T> = GF(3^21)
-    sage: f = FiniteFieldHomomorphism_generic(Hom(k, K)); f
+    sage: f = FiniteFieldHomomorphism_generic(Hom(k, K)); f # random
     Ring morphism:
       From: Finite Field in t of size 3^7
       To:   Finite Field in T of size 3^21
       Defn: t |--> T^20 + 2*T^18 + T^16 + 2*T^13 + T^9 + 2*T^8 + T^7 + T^6 + T^5 + T^3 + 2*T^2 + T
 
-    sage: f(t)
+    sage: f(t) # random
     T^20 + 2*T^18 + T^16 + 2*T^13 + T^9 + 2*T^8 + T^7 + T^6 + T^5 + T^3 + 2*T^2 + T
+    sage: f(t) == f.im_gens()[0]
+    True
 
 The map `f` has a method ``section`` which returns a partially defined
 map which is the inverse of `f` on the image of `f`::
 
-    sage: g = f.section(); g
+    sage: g = f.section(); g # random
     Section of Ring morphism:
       From: Finite Field in t of size 3^7
       To:   Finite Field in T of size 3^21
       Defn: t |--> T^20 + 2*T^18 + T^16 + 2*T^13 + T^9 + 2*T^8 + T^7 + T^6 + T^5 + T^3 + 2*T^2 + T
-    sage: g(f(t^3+t^2+1))
-    t^3 + t^2 + 1
+    sage: a = k.random_element()
+    sage: g(f(a)) == a
+    True
     sage: g(T)
     Traceback (most recent call last):
     ...
     ValueError: T is not in the image of Ring morphism:
       From: Finite Field in t of size 3^7
       To:   Finite Field in T of size 3^21
-      Defn: t |--> T^20 + 2*T^18 + T^16 + 2*T^13 + T^9 + 2*T^8 + T^7 + T^6 + T^5 + T^3 + 2*T^2 + T
+      Defn: ...
 
 There is no embedding of `GF(5^6)` into `GF(5^11)`::
 
@@ -130,16 +133,17 @@ cdef class SectionFiniteFieldHomomorphism_generic(Section):
             sage: K.<T> = GF(3^21)
             sage: f = FiniteFieldHomomorphism_generic(Hom(k, K))
             sage: g = f.section()
-            sage: g(f(t^3+t^2+1))
-            t^3 + t^2 + 1
+            sage: a = k.random_element()
+            sage: g(f(a)) == a
+            True
 
             sage: g(T)
             Traceback (most recent call last):
             ...
             ValueError: T is not in the image of Ring morphism:
               From: Finite Field in t of size 3^7
               To:   Finite Field in T of size 3^21
-              Defn: t |--> T^20 + 2*T^18 + T^16 + 2*T^13 + T^9 + 2*T^8 + T^7 + T^6 + T^5 + T^3 + 2*T^2 + T
+              Defn: t |--> ...
         """
         for root, _ in x.minimal_polynomial().roots(ring=self.codomain()):
             if self._inverse(root) == x:
@@ -158,7 +162,7 @@ cdef class SectionFiniteFieldHomomorphism_generic(Section):
             sage: K.<T> = GF(3^21)
             sage: f = FiniteFieldHomomorphism_generic(Hom(k, K))
             sage: g = f.section()
-            sage: g._repr_()
+            sage: g._repr_() # random
             'Section of Ring morphism:\n  From: Finite Field in t of size 3^7\n  To:   Finite Field in T of size 3^21\n  Defn: t |--> T^20 + 2*T^18 + T^16 + 2*T^13 + T^9 + 2*T^8 + T^7 + T^6 + T^5 + T^3 + 2*T^2 + T'
         """
         return "Section of %s" % self._inverse
@@ -202,11 +206,15 @@ cdef class FiniteFieldHomomorphism_generic(RingHomomorphism_im_gens):
             sage: from sage.rings.finite_rings.hom_finite_field import FiniteFieldHomomorphism_generic
             sage: k.<t> = GF(3^7)
             sage: K.<T> = GF(3^21)
-            sage: f = FiniteFieldHomomorphism_generic(Hom(k, K)); f
+            sage: f = FiniteFieldHomomorphism_generic(Hom(k, K)); f # random
             Ring morphism:
               From: Finite Field in t of size 3^7
               To:   Finite Field in T of size 3^21
               Defn: t |--> T^20 + 2*T^18 + T^16 + 2*T^13 + T^9 + 2*T^8 + T^7 + T^6 + T^5 + T^3 + 2*T^2 + T
+            sage: a = k.random_element()
+            sage: b = k.random_element()
+            sage: f(a) + f(b) == f(a + b)
+            True
 
             sage: k.<t> = GF(3^6)
             sage: K.<t> = GF(3^9)
@@ -299,7 +307,7 @@ cdef class FiniteFieldHomomorphism_generic(RingHomomorphism_im_gens):
             sage: k.<t> = GF(3^3)
             sage: K.<T> = GF(3^9)
             sage: f = FiniteFieldHomomorphism_generic(Hom(k, K))
-            sage: f(t)
+            sage: f(t) # random
             2*T^6 + 2*T^4 + T^2 + T
 
             sage: a = k.random_element()
@@ -368,20 +376,22 @@ cdef class FiniteFieldHomomorphism_generic(RingHomomorphism_im_gens):
             sage: k.<t> = GF(3^7)
             sage: K.<T> = GF(3^21)
             sage: f = FiniteFieldHomomorphism_generic(Hom(k, K))
-            sage: g = f.section(); g
+            sage: g = f.section(); g # random
             Section of Ring morphism:
               From: Finite Field in t of size 3^7
               To:   Finite Field in T of size 3^21
               Defn: t |--> T^20 + 2*T^18 + T^16 + 2*T^13 + T^9 + 2*T^8 + T^7 + T^6 + T^5 + T^3 + 2*T^2 + T
-            sage: g(f(t^3+t^2+1))
-            t^3 + t^2 + 1
+            sage: a = k.random_element()
+            sage: b = k.random_element()
+            sage: g(f(a) + f(b)) == a + b
+            True
             sage: g(T)
             Traceback (most recent call last):
             ...
             ValueError: T is not in the image of Ring morphism:
               From: Finite Field in t of size 3^7
               To:   Finite Field in T of size 3^21
-              Defn: t |--> T^20 + 2*T^18 + T^16 + 2*T^13 + T^9 + 2*T^8 + T^7 + T^6 + T^5 + T^3 + 2*T^2 + T
+              Defn: ...
         """
         if self.base_map() is not None:
             raise NotImplementedError
@@ -421,7 +431,7 @@ cdef class FiniteFieldHomomorphism_generic(RingHomomorphism_im_gens):
             sage: k.<t> = GF(5^3)
             sage: Frob = k.frobenius_endomorphism()
             sage: embed = Frob.fixed_field()[1]
-            sage: hash(embed)  # random
+            sage: hash(embed) # random
             -2441354824160407762
         """
         return Morphism.__hash__(self)
@@ -751,15 +761,17 @@ cdef class FrobeniusEndomorphism_finite_field(FrobeniusEndomorphism_generic):
             sage: kfixed, embed = f.fixed_field()
             sage: kfixed
             Finite Field in t_fixed of size 5^2
-            sage: embed
+            sage: embed # random
             Ring morphism:
               From: Finite Field in t_fixed of size 5^2
               To:   Finite Field in t of size 5^6
               Defn: t_fixed |--> 4*t^5 + 2*t^4 + 4*t^2 + t
 
             sage: tfixed = kfixed.gen()
-            sage: embed(tfixed)
+            sage: embed(tfixed) # random
             4*t^5 + 2*t^4 + 4*t^2 + t
+            sage: embed(tfixed) == embed.im_gens()[0]
+            True
         """
         if self._degree_fixed == 1:
             k = FiniteField(self.domain().characteristic())

src/sage/rings/finite_rings/hom_finite_field_givaro.pyx

@@ -59,11 +59,15 @@ cdef class SectionFiniteFieldHomomorphism_givaro(SectionFiniteFieldHomomorphism_
             sage: k.<t> = GF(3^2)
             sage: K.<T> = GF(3^4)
             sage: f = FiniteFieldHomomorphism_givaro(Hom(k, K))
-            sage: g = f.section(); g
+            sage: g = f.section(); g # random
             Section of Ring morphism:
               From: Finite Field in t of size 3^2
               To:   Finite Field in T of size 3^4
               Defn: t |--> 2*T^3 + 2*T^2 + 1
+            sage: a = k.random_element()
+            sage: b = k.random_element()
+            sage: g(f(a) + f(b)) == g(f(a)) + g(f(b)) == a + b
+            True
         """
         if not isinstance(inverse, FiniteFieldHomomorphism_givaro):
             raise TypeError("The given map is not an instance of FiniteFieldHomomorphism_givaro")
@@ -114,7 +118,7 @@ cdef class SectionFiniteFieldHomomorphism_givaro(SectionFiniteFieldHomomorphism_
             ValueError: T is not in the image of Ring morphism:
               From: Finite Field in t of size 3^2
               To:   Finite Field in T of size 3^4
-              Defn: t |--> 2*T^3 + 2*T^2 + 1
+              Defn: t |--> ...
         """
         if x.parent() != self.domain():
             raise TypeError("%s is not in %s" % (x, self.domain()))
@@ -140,11 +144,15 @@ cdef class FiniteFieldHomomorphism_givaro(FiniteFieldHomomorphism_generic):
             sage: from sage.rings.finite_rings.hom_finite_field_givaro import FiniteFieldHomomorphism_givaro
             sage: k.<t> = GF(3^2)
             sage: K.<T> = GF(3^4)
-            sage: f = FiniteFieldHomomorphism_givaro(Hom(k, K)); f
+            sage: f = FiniteFieldHomomorphism_givaro(Hom(k, K)); f # random
             Ring morphism:
               From: Finite Field in t of size 3^2
               To:   Finite Field in T of size 3^4
               Defn: t |--> 2*T^3 + 2*T^2 + 1
+            sage: a = k.random_element()
+            sage: b = k.random_element()
+            sage: f(a) + f(b) == f(a + b)
+            True
 
             sage: k.<t> = GF(3^10)
             sage: K.<T> = GF(3^20)
@@ -182,8 +190,10 @@ cdef class FiniteFieldHomomorphism_givaro(FiniteFieldHomomorphism_generic):
             sage: k.<t> = GF(3^2)
             sage: K.<T> = GF(3^4)
             sage: f = FiniteFieldHomomorphism_givaro(Hom(k, K))
-            sage: f(t)
+            sage: f(t) # random
             2*T^3 + 2*T^2 + 1
+            sage: f(t) == f.im_gens()[0]
+            True
         """
         if x.parent() != self.domain():
             raise TypeError("%s is not in %s" % (x, self.domain()))
@@ -242,14 +252,14 @@ cdef class FrobeniusEndomorphism_givaro(FrobeniusEndomorphism_finite_field):
             sage: kfixed, embed = f.fixed_field()
             sage: kfixed
             Finite Field in t_fixed of size 5^2
-            sage: embed
+            sage: embed # random
             Ring morphism:
               From: Finite Field in t_fixed of size 5^2
               To:   Finite Field in t of size 5^6
               Defn: t_fixed |--> 4*t^5 + 2*t^4 + 4*t^2 + t
 
             sage: tfixed = kfixed.gen()
-            sage: embed(tfixed)
+            sage: embed(tfixed) # random
             4*t^5 + 2*t^4 + 4*t^2 + t
         """
         if self._degree_fixed == 1:

src/sage/rings/finite_rings/homset.py

@@ -334,7 +334,7 @@ def _an_element_(self):
 
         TESTS::
 
-            sage: Hom(GF(3^3, 'a'), GF(3^6, 'b')).an_element()
+            sage: Hom(GF(3^3, 'a'), GF(3^6, 'b')).an_element()  # random
             Ring morphism:
               From: Finite Field in a of size 3^3
               To:   Finite Field in b of size 3^6