diff --git a/src/sage/rings/complex_arb.pyx b/src/sage/rings/complex_arb.pyx
index 70a5e3f2423..095de7b6774 100644
--- a/src/sage/rings/complex_arb.pyx
+++ b/src/sage/rings/complex_arb.pyx
@@ -1261,6 +1261,7 @@ class ComplexBallField(UniqueRepresentation, sage.rings.abc.ComplexBallField):
 
         return res
 
+
 cdef inline bint _do_sig(long prec) noexcept:
     """
     Whether signal handlers should be installed for calls to FLINT.
diff --git a/src/sage/rings/complex_double.pyx b/src/sage/rings/complex_double.pyx
index aa4e284f231..460d546b200 100644
--- a/src/sage/rings/complex_double.pyx
+++ b/src/sage/rings/complex_double.pyx
@@ -505,7 +505,6 @@ cdef class ComplexDoubleField_class(sage.rings.abc.ComplexDoubleField):
             from sage.rings.complex_mpfr import ComplexField
             return ComplexField(prec)
 
-
     def gen(self, n=0):
         """
         Return the generator of the complex double field.
@@ -699,6 +698,7 @@ cdef ComplexDoubleElement new_ComplexDoubleElement():
     z = ComplexDoubleElement.__new__(ComplexDoubleElement)
     return z
 
+
 def is_ComplexDoubleElement(x):
     """
     Return ``True`` if ``x`` is a :class:`ComplexDoubleElement`.
@@ -1530,7 +1530,6 @@ cdef class ComplexDoubleElement(FieldElement):
         else:
             return z
 
-
     def is_square(self):
         r"""
         This function always returns ``True`` as `\CC` is algebraically closed.
@@ -2018,7 +2017,6 @@ cdef class ComplexDoubleElement(FieldElement):
         """
         return self._new_c(gsl_complex_tanh(self._complex))
 
-
     def sech(self):
         r"""
         This function returns the complex hyperbolic secant of the complex
@@ -2594,8 +2592,6 @@ cdef class ComplexToCDF(Morphism):
         return "Native"
 
 
-
-
 #####################################################
 # unique objects
 #####################################################
@@ -2604,6 +2600,7 @@ _CDF = ComplexDoubleField_class()
 CDF = _CDF  # external interface
 cdef ComplexDoubleElement I = ComplexDoubleElement(0,1)
 
+
 def ComplexDoubleField():
     """
     Returns the field of double precision complex numbers.
@@ -2617,6 +2614,7 @@ def ComplexDoubleField():
     """
     return _CDF
 
+
 from sage.misc.parser import Parser
 cdef cdf_parser = Parser(float, float,  {"I" : _CDF.gen(), "i" : _CDF.gen()})
 
diff --git a/src/sage/rings/complex_mpc.pyx b/src/sage/rings/complex_mpc.pyx
index 1a4c49504c0..eda06a9fc2c 100644
--- a/src/sage/rings/complex_mpc.pyx
+++ b/src/sage/rings/complex_mpc.pyx
@@ -95,6 +95,7 @@ AA = None
 QQbar = None
 CDF = CLF = RLF = None
 
+
 def late_import():
     """
     Import the objects/modules after build (when needed).
@@ -112,6 +113,7 @@ def late_import():
         from sage.rings.real_lazy import CLF, RLF
         from sage.rings.complex_double import CDF
 
+
 _mpfr_rounding_modes = ['RNDN', 'RNDZ', 'RNDU', 'RNDD']
 
 _mpc_rounding_modes = [ 'RNDNN', 'RNDZN', 'RNDUN', 'RNDDN',
@@ -233,6 +235,8 @@ cpdef inline split_complex_string(string, int base=10):
 # their parent via direct C calls, which will be faster.
 
 cache = {}
+
+
 def MPComplexField(prec=53, rnd="RNDNN", names=None):
     """
     Return the complex field with real and imaginary parts having
@@ -2412,6 +2416,7 @@ cdef inline mp_exp_t max_exp(MPComplexNumber z) noexcept:
         return mpfr_get_exp(z.value.im)
     return max_exp_t(mpfr_get_exp(z.value.re), mpfr_get_exp(z.value.im))
 
+
 def __create__MPComplexField_version0 (prec, rnd):
     """
     Create a :class:`MPComplexField`.
@@ -2424,6 +2429,7 @@ def __create__MPComplexField_version0 (prec, rnd):
     """
     return MPComplexField(prec, rnd)
 
+
 def __create__MPComplexNumber_version0 (parent, s, base=10):
     """
     Create a :class:`MPComplexNumber`.
@@ -2439,6 +2445,7 @@ def __create__MPComplexNumber_version0 (parent, s, base=10):
     """
     return MPComplexNumber(parent, s, base=base)
 
+
 # original version of the file had this with only 1 underscore - TCS
 __create_MPComplexNumber_version0 = __create__MPComplexNumber_version0
 
diff --git a/src/sage/rings/complex_mpfr.pyx b/src/sage/rings/complex_mpfr.pyx
index 6022d4147b9..e1ce333d57d 100644
--- a/src/sage/rings/complex_mpfr.pyx
+++ b/src/sage/rings/complex_mpfr.pyx
@@ -69,6 +69,8 @@ NumberFieldElement_quadratic = ()
 AA = None
 QQbar = None
 CDF = CLF = RLF = None
+
+
 def late_import():
     """
     Import the objects/modules after build (when needed).
@@ -89,12 +91,14 @@ def late_import():
         from sage.rings.real_lazy import CLF, RLF
         from sage.rings.complex_double import CDF
 
+
 cdef object numpy_complex_interface = {'typestr': '=c16'}
 cdef object numpy_object_interface = {'typestr': '|O'}
 
 cdef mpfr_rnd_t rnd
 rnd = MPFR_RNDN
 
+
 def set_global_complex_round_mode(n):
     """
     Set the global complex rounding mode.
@@ -111,6 +115,7 @@ def set_global_complex_round_mode(n):
     global rnd
     rnd = n
 
+
 def is_ComplexNumber(x):
     r"""
     Return ``True`` if ``x`` is a complex number. In particular, if ``x`` is
@@ -145,6 +150,8 @@ def is_ComplexNumber(x):
 
 
 cache = {}
+
+
 def ComplexField(prec=53, names=None):
     """
     Return the complex field with real and imaginary parts having prec
@@ -673,7 +680,6 @@ class ComplexField_class(sage.rings.abc.ComplexField):
         from sage.categories.pushout import AlgebraicClosureFunctor
         return (AlgebraicClosureFunctor(), self._real_field())
 
-
     def random_element(self, component_max=1, *args, **kwds):
         r"""
         Return a uniformly distributed random number inside a square
@@ -857,6 +863,7 @@ class ComplexField_class(sage.rings.abc.ComplexField):
         return Factorization([(R(gg).monic(), e) for gg, e in zip(*F)],
                              f.leading_coefficient())
 
+
 cdef class ComplexNumber(sage.structure.element.FieldElement):
     """
     A floating point approximation to a complex number using any
@@ -1429,7 +1436,6 @@ cdef class ComplexNumber(sage.structure.element.FieldElement):
         """
         return gmpy2.GMPy_MPC_From_mpfr(self.__re, self.__im)
 
-
     def _mpmath_(self, prec=None, rounding=None):
         """
         Return an mpmath version of ``self``.
@@ -2300,8 +2306,6 @@ cdef class ComplexNumber(sage.structure.element.FieldElement):
         mpfr_clear(ch)
         return z
 
-
-
     def eta(self, omit_frac=False):
         r"""
         Return the value of the Dedekind `\eta` function on ``self``,
@@ -2367,7 +2371,6 @@ cdef class ComplexNumber(sage.structure.element.FieldElement):
         except sage.libs.pari.all.PariError:
             raise ValueError("value must be in the upper half plane")
 
-
     def sin(self):
         """
         Return the sine of ``self``.
@@ -2461,7 +2464,6 @@ cdef class ComplexNumber(sage.structure.element.FieldElement):
         mpfr_clear(a)
         return z
 
-
     def tanh(self):
         """
         Return the hyperbolic tangent of ``self``.
@@ -2729,7 +2731,6 @@ cdef class ComplexNumber(sage.structure.element.FieldElement):
         mpfr_atan2(x.value, self.__im, self.__re, rnd)
         return x
 
-
     def arg(self):
         """
         See :meth:`argument`.
@@ -3080,7 +3081,6 @@ cdef class ComplexNumber(sage.structure.element.FieldElement):
         mpfr_clear(r)
         return zlist
 
-
     def is_square(self):
         r"""
         This function always returns true as `\CC` is algebraically closed.
@@ -3498,6 +3498,7 @@ cpdef int cmp_abs(ComplexNumber a, ComplexNumber b) noexcept:
 
     return res
 
+
 def _format_complex_number(real, imag, format_spec):
     """
     Construct a formatted string from real and imaginary parts.
diff --git a/src/sage/rings/factorint.pyx b/src/sage/rings/factorint.pyx
index 2144e6d71c7..5e8ae9ddd94 100644
--- a/src/sage/rings/factorint.pyx
+++ b/src/sage/rings/factorint.pyx
@@ -208,6 +208,7 @@ cpdef factor_aurifeuillian(n, check=True):
             return F
     return [n]
 
+
 def factor_cunningham(m, proof=None):
     r"""
     Return factorization of ``self`` obtained using trial division
@@ -251,6 +252,7 @@ def factor_cunningham(m, proof=None):
     else:
         return IntegerFactorization(L)*n.factor(proof=proof)
 
+
 cpdef factor_trial_division(m, long limit=LONG_MAX):
     r"""
     Return partial factorization of ``self`` obtained using trial division
diff --git a/src/sage/rings/finite_rings/element_givaro.pyx b/src/sage/rings/finite_rings/element_givaro.pyx
index c9b9e323181..c1ab2de2aaf 100644
--- a/src/sage/rings/finite_rings/element_givaro.pyx
+++ b/src/sage/rings/finite_rings/element_givaro.pyx
@@ -1740,6 +1740,7 @@ def unpickle_FiniteField_givaroElement(parent, int x):
     """
     return make_FiniteField_givaroElement(parent._cache, x)
 
+
 from sage.misc.persist import register_unpickle_override
 register_unpickle_override('sage.rings.finite_field_givaro', 'unpickle_FiniteField_givaroElement', unpickle_FiniteField_givaroElement)
 
diff --git a/src/sage/rings/finite_rings/element_ntl_gf2e.pyx b/src/sage/rings/finite_rings/element_ntl_gf2e.pyx
index 399bb76815b..512ac4ec1d2 100644
--- a/src/sage/rings/finite_rings/element_ntl_gf2e.pyx
+++ b/src/sage/rings/finite_rings/element_ntl_gf2e.pyx
@@ -1314,5 +1314,6 @@ def unpickleFiniteField_ntl_gf2eElement(parent, elem):
     """
     return parent(elem)
 
+
 from sage.misc.persist import register_unpickle_override
 register_unpickle_override('sage.rings.finite_field_ntl_gf2e', 'unpickleFiniteField_ntl_gf2eElement', unpickleFiniteField_ntl_gf2eElement)
diff --git a/src/sage/rings/finite_rings/finite_field_base.pyx b/src/sage/rings/finite_rings/finite_field_base.pyx
index 661b2ba08e6..8be832999b2 100644
--- a/src/sage/rings/finite_rings/finite_field_base.pyx
+++ b/src/sage/rings/finite_rings/finite_field_base.pyx
@@ -277,7 +277,6 @@ cdef class FiniteField(Field):
         return "GF(%s,Variable=>symbol %s)" % (self.order(),
                                                self.variable_name())
 
-
     def _sage_input_(self, sib, coerced):
         r"""
         Produce an expression which will reproduce this value when evaluated.
@@ -1115,7 +1114,6 @@ cdef class FiniteField(Field):
         """
         return self.order() - 1
 
-
     def random_element(self, *args, **kwds):
         r"""
         A random element of the finite field.  Passes arguments to
@@ -2162,6 +2160,7 @@ cdef class FiniteField(Field):
         python_int = int.from_bytes(input_bytes, byteorder=byteorder)
         return self.from_integer(python_int)
 
+
 def unpickle_FiniteField_ext(_type, order, variable_name, modulus, kwargs):
     r"""
     Used to unpickle extensions of finite fields. Now superseded (hence no
@@ -2177,6 +2176,7 @@ def unpickle_FiniteField_prm(_type, order, variable_name, kwargs):
     """
     return _type(order, variable_name, **kwargs)
 
+
 register_unpickle_override(
     'sage.rings.ring', 'unpickle_FiniteField_prm', unpickle_FiniteField_prm)
 
diff --git a/src/sage/rings/finite_rings/integer_mod.pyx b/src/sage/rings/finite_rings/integer_mod.pyx
index 94a82bc3d47..097f8ac87d3 100644
--- a/src/sage/rings/finite_rings/integer_mod.pyx
+++ b/src/sage/rings/finite_rings/integer_mod.pyx
@@ -163,6 +163,7 @@ mod = Mod
 register_unpickle_override('sage.rings.integer_mod', 'Mod', Mod)
 register_unpickle_override('sage.rings.integer_mod', 'mod', mod)
 
+
 def IntegerMod(parent, value):
     """
     Create an integer modulo `n` with the given parent.
@@ -3462,7 +3463,6 @@ cdef class IntegerMod_int64(IntegerMod_abstract):
         """
         return self._new_c((self.ivalue * (<IntegerMod_int64>right).ivalue) % self._modulus.int64)
 
-
     cpdef _div_(self, right):
         """
         EXAMPLES::
@@ -3995,6 +3995,7 @@ def square_root_mod_prime_power(IntegerMod_abstract a, p, e):
         x *= p**(val//2)
     return x
 
+
 cpdef square_root_mod_prime(IntegerMod_abstract a, p=None):
     r"""
     Calculates the square root of `a`, where `a` is an
diff --git a/src/sage/rings/finite_rings/residue_field.pyx b/src/sage/rings/finite_rings/residue_field.pyx
index e4e617a33eb..5e9d74844e3 100644
--- a/src/sage/rings/finite_rings/residue_field.pyx
+++ b/src/sage/rings/finite_rings/residue_field.pyx
@@ -919,6 +919,7 @@ class ResidueField_generic(Field):
         """
         return 1 + hash(self.ideal())
 
+
 cdef class ReductionMap(Map):
     """
     A reduction map from a (subset) of a number field or function field to
@@ -1685,6 +1686,7 @@ cdef class LiftingMap(Section):
         """
         return "Lifting"
 
+
 class ResidueFiniteField_prime_modn(ResidueField_generic, FiniteField_prime_modn):
     """
     The class representing residue fields of number fields that have
diff --git a/src/sage/rings/function_field/element.pyx b/src/sage/rings/function_field/element.pyx
index 2765b805aba..4c4719f0f31 100644
--- a/src/sage/rings/function_field/element.pyx
+++ b/src/sage/rings/function_field/element.pyx
@@ -86,6 +86,7 @@ def is_FunctionFieldElement(x):
     from sage.rings.function_field.function_field import is_FunctionField
     return is_FunctionField(x.parent())
 
+
 def make_FunctionFieldElement(parent, element_class, representing_element):
     """
     Used for unpickling FunctionFieldElement objects (and subclasses).
diff --git a/src/sage/rings/function_field/hermite_form_polynomial.pyx b/src/sage/rings/function_field/hermite_form_polynomial.pyx
index 12881615315..2aa4516000c 100644
--- a/src/sage/rings/function_field/hermite_form_polynomial.pyx
+++ b/src/sage/rings/function_field/hermite_form_polynomial.pyx
@@ -49,13 +49,14 @@ AUTHORS:
 #  Distributed under the terms of the GNU General Public License (GPL)
 #  as published by the Free Software Foundation; either version 2 of
 #  the License, or (at your option) any later version.
-#                  http://www.gnu.org/licenses/
+#                  https://www.gnu.org/licenses/
 # ****************************************************************************
 
 from sage.matrix.matrix cimport Matrix
 from sage.rings.polynomial.polynomial_element cimport Polynomial
 from sage.matrix.constructor import identity_matrix
 
+
 def reversed_hermite_form(Matrix mat, bint transformation=False):
     """
     Transform the matrix in place to reversed hermite normal form and
diff --git a/src/sage/rings/integer_ring.pyx b/src/sage/rings/integer_ring.pyx
index d0fb7ac3bc5..4c384622ff9 100644
--- a/src/sage/rings/integer_ring.pyx
+++ b/src/sage/rings/integer_ring.pyx
@@ -87,6 +87,7 @@ cdef int number_of_integer_rings = 0
 #   sigma, we do not recreate the sampler but take it from this "cache".
 _prev_discrete_gaussian_integer_sampler = (None, None)
 
+
 def is_IntegerRing(x):
     r"""
     Internal function: return ``True`` iff ``x`` is the ring `\ZZ` of integers.
@@ -105,6 +106,7 @@ def is_IntegerRing(x):
     """
     return isinstance(x, IntegerRing_class)
 
+
 cdef class IntegerRing_class(CommutativeRing):
     r"""
     The ring of integers.
@@ -1406,7 +1408,6 @@ cdef class IntegerRing_class(CommutativeRing):
 
         g = g.gcd(R( {e[j] - e[i_min]: c[j] for j in range(i_min, k)} ))
 
-
         cdef list cc
         cdef list ee
         cdef int m1, m2
@@ -1609,6 +1610,7 @@ cdef class IntegerRing_class(CommutativeRing):
 ZZ = IntegerRing_class()
 Z = ZZ
 
+
 def IntegerRing():
     """
     Return the integer ring.
@@ -1622,6 +1624,7 @@ def IntegerRing():
     """
     return ZZ
 
+
 def crt_basis(X, xgcd=None):
     r"""
     Compute and return a Chinese Remainder Theorem basis for the list ``X``
diff --git a/src/sage/rings/number_field/number_field_element_quadratic.pyx b/src/sage/rings/number_field/number_field_element_quadratic.pyx
index f48df864b74..54d8c9cc329 100644
--- a/src/sage/rings/number_field/number_field_element_quadratic.pyx
+++ b/src/sage/rings/number_field/number_field_element_quadratic.pyx
@@ -79,6 +79,7 @@ def __make_NumberFieldElement_quadratic0(parent, a, b, denom):
     """
     return NumberFieldElement_quadratic(parent, (a, b, denom))
 
+
 def __make_NumberFieldElement_quadratic1(parent, cls, a, b, denom):
     """
     Used in unpickling elements of number fields.
@@ -410,13 +411,11 @@ cdef class NumberFieldElement_quadratic(NumberFieldElement_absolute):
         mpz_init(self.b)
         mpz_init(self.denom)
 
-
     def __dealloc__(self):
         mpz_clear(self.a)
         mpz_clear(self.b)
         mpz_clear(self.denom)
 
-
     def __reduce__(self):
         """
         Used for pickling.
@@ -472,7 +471,6 @@ cdef class NumberFieldElement_quadratic(NumberFieldElement_absolute):
         self._reduce_c_()
         return 0  # No error
 
-
     def _lift_cyclotomic_element(self, new_parent, bint check=True, int rel=0):
         """
         Creates an element of the passed field from this field.  This
@@ -1317,7 +1315,6 @@ cdef class NumberFieldElement_quadratic(NumberFieldElement_absolute):
             mpz_neg(self.b, self.b)
         mpz_clear(gcd)
 
-
     cpdef _add_(self, other_m):
         """
         EXAMPLES::
@@ -1375,7 +1372,6 @@ cdef class NumberFieldElement_quadratic(NumberFieldElement_absolute):
         res._reduce_c_()
         return res
 
-
     cpdef _sub_(self, other_m):
         """
         EXAMPLES::
@@ -1423,7 +1419,6 @@ cdef class NumberFieldElement_quadratic(NumberFieldElement_absolute):
         res._reduce_c_()
         return res
 
-
     def __neg__(self):
         """
         EXAMPLES::
diff --git a/src/sage/rings/number_field/number_field_morphisms.pyx b/src/sage/rings/number_field/number_field_morphisms.pyx
index e61b63117f1..9ef71586355 100644
--- a/src/sage/rings/number_field/number_field_morphisms.pyx
+++ b/src/sage/rings/number_field/number_field_morphisms.pyx
@@ -444,6 +444,7 @@ cpdef closest(target, values, margin=1):
         else:
             return None
 
+
 def root_from_approx(f, a):
     """
     Return an exact root of the polynomial `f` closest to `a`.
@@ -505,6 +506,7 @@ def root_from_approx(f, a):
             raise ValueError("{} is not a root of {}".format(a, f))
         return a
 
+
 def create_embedding_from_approx(K, gen_image):
     """
     Return an embedding of ``K`` determined by ``gen_image``.
diff --git a/src/sage/rings/number_field/totallyreal.pyx b/src/sage/rings/number_field/totallyreal.pyx
index f50de987968..624e04c1641 100644
--- a/src/sage/rings/number_field/totallyreal.pyx
+++ b/src/sage/rings/number_field/totallyreal.pyx
@@ -466,6 +466,7 @@ def enumerate_totallyreal_fields_prim(n, B, a = [], verbose=0, return_seqs=False
         return [[ZZ(s[0]), Px([QQ(x) for x in s[1].list()])]
                 for s in S]
 
+
 def weed_fields(S, Py_ssize_t lenS=0):
     r"""
     Function used internally by the :func:`~enumerate_totallyreal_fields_prim`
diff --git a/src/sage/rings/number_field/totallyreal_data.pyx b/src/sage/rings/number_field/totallyreal_data.pyx
index 32efb720668..e02a3074cce 100644
--- a/src/sage/rings/number_field/totallyreal_data.pyx
+++ b/src/sage/rings/number_field/totallyreal_data.pyx
@@ -109,6 +109,7 @@ def hermite_constant(n):
 
     return gamma
 
+
 cdef double eval_seq_as_poly(int *f, int n, double x) noexcept:
     r"""
     Evaluates the sequence a, thought of as a polynomial with
@@ -291,11 +292,10 @@ cpdef lagrange_degree_3(int n, int an1, int an2, int an3):
                     3*nrsq*s1sq*s2sq - 3*nr*s1fo*s2 + \
                     s1sq*s1fo
 
-
         fcoeff = [ int(coeffs[i]) for i in range(7) ]
         f = ZZx(fcoeff)
         df = ZZx([i*coeffs[i] for i in range(1,7)])
-        f = f//gcd(f,df)
+        f = f//gcd(f, df)
         fcoeff = [int(c) for c in f.list()]
 
         rts = RRx(fcoeff).roots()
@@ -315,6 +315,7 @@ primessq_py = [4, 9, 25, 49, 121, 169, 289, 361, 529, 841, 961, 1369, 1681, 1849
 for i from 0 <= i < 46:
     primessq[i] = primessq_py[i]
 
+
 def int_has_small_square_divisor(sage.rings.integer.Integer d):
     r"""
     Return the largest `a` such that `a^2` divides `d` and `a` has prime divisors `< 200`.
@@ -343,6 +344,7 @@ def int_has_small_square_divisor(sage.rings.integer.Integer d):
 
     return asq
 
+
 cdef int eval_seq_as_poly_int(int *f, int n, int x) noexcept:
     r"""
     Evaluates the sequence a, thought of as a polynomial with
@@ -406,6 +408,7 @@ cdef int easy_is_irreducible(int *a, int n) noexcept:
 
     return 1
 
+
 def easy_is_irreducible_py(f):
     """
     Used solely for testing easy_is_irreducible.
@@ -424,7 +427,6 @@ def easy_is_irreducible_py(f):
     return easy_is_irreducible(a, len(f)-1)
 
 
-
 #****************************************************************************
 # Main class and routine
 #****************************************************************************
@@ -512,7 +514,6 @@ cdef class tr_data:
             self.beta[i] = <double>0
             self.gnk[i] = 0
 
-
         # Initialize variables.
         if a == []:
             # No starting input, all polynomials will be found; initialize to zero.
@@ -841,7 +842,6 @@ cdef class tr_data:
                         if tmp_dbl > akmin:
                             akmin = tmp_dbl
 
-
                     akmax = -eval_seq_as_poly(gnkm, n-k, betak[nk]) \
                             +fabs(eval_seq_as_poly(gnkm1, n-(k+1), betak[nk]))*eps_global
                     for i from 1 <= i < nk/2+1:
diff --git a/src/sage/rings/padics/padic_ZZ_pX_element.pyx b/src/sage/rings/padics/padic_ZZ_pX_element.pyx
index b88bd5053bf..bf892956a16 100644
--- a/src/sage/rings/padics/padic_ZZ_pX_element.pyx
+++ b/src/sage/rings/padics/padic_ZZ_pX_element.pyx
@@ -738,6 +738,7 @@ def _find_val_aprec_test(R, L):
     """
     return find_val_aprec(R.prime_pow, L)
 
+
 cdef find_val_aprec(PowComputer_ext pp, L):
     r"""
     Given a list ``L``, finds the minimum valuation, minimum absolute
@@ -843,6 +844,7 @@ def _test_get_val_prec(R, a):
     """
     return get_val_prec(R.prime_pow, a)
 
+
 cdef get_val_prec(PowComputer_ext pp, a):
     r"""
     Return valuation, absolute precision and type of an input element.
diff --git a/src/sage/rings/padics/padic_generic_element.pyx b/src/sage/rings/padics/padic_generic_element.pyx
index f7a7ab19ceb..145371c28c9 100644
--- a/src/sage/rings/padics/padic_generic_element.pyx
+++ b/src/sage/rings/padics/padic_generic_element.pyx
@@ -4515,6 +4515,7 @@ def _compute_g(p, n, prec, terms):
         g[i+1] = -(g[i]/(v-v**2)).integral()
     return [x.truncate(terms) for x in g]
 
+
 cpdef dwork_mahler_coeffs(R, int bd=20):
     r"""
     Compute Dwork's formula for Mahler coefficients of `p`-adic Gamma.
diff --git a/src/sage/rings/padics/padic_printing.pyx b/src/sage/rings/padics/padic_printing.pyx
index fc9f15b07b1..f134c5afc45 100644
--- a/src/sage/rings/padics/padic_printing.pyx
+++ b/src/sage/rings/padics/padic_printing.pyx
@@ -294,6 +294,7 @@ class pAdicPrinterDefaults(SageObject):
         else:
             self._alphabet = tuple(alphabet)
 
+
 _printer_defaults = pAdicPrinterDefaults()
 
 
diff --git a/src/sage/rings/padics/pow_computer_ext.pyx b/src/sage/rings/padics/pow_computer_ext.pyx
index 89a073550f5..8ce4953fadf 100644
--- a/src/sage/rings/padics/pow_computer_ext.pyx
+++ b/src/sage/rings/padics/pow_computer_ext.pyx
@@ -298,6 +298,7 @@ def ZZ_pX_eis_shift_test(_shifter, _a, _n, _finalprec):
     ZZ_pX_eis_shift_p(shifter, &x.x, &a.x, n, finalprec)
     return x
 
+
 cdef int ZZ_pX_eis_shift_p(PowComputer_ZZ_pX self, ZZ_pX_c* x, ZZ_pX_c* a, long n, long finalprec) except -1:
     """
     Eis-shifts a over by n and puts the result into x.
diff --git a/src/sage/rings/polynomial/multi_polynomial.pyx b/src/sage/rings/polynomial/multi_polynomial.pyx
index 6a30e521302..2f4a33dc358 100644
--- a/src/sage/rings/polynomial/multi_polynomial.pyx
+++ b/src/sage/rings/polynomial/multi_polynomial.pyx
@@ -20,12 +20,14 @@ from itertools import chain
 
 from sage.misc.misc_c import prod
 
+
 def is_MPolynomial(x):
     from sage.misc.superseded import deprecation
     deprecation(32709, "the function is_MPolynomial is deprecated; use isinstance(x, sage.rings.polynomial.multi_polynomial.MPolynomial) instead")
 
     return isinstance(x, MPolynomial)
 
+
 from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
 from sage.categories.map cimport Map
 from sage.rings.rational_field import QQ
diff --git a/src/sage/rings/polynomial/polynomial_modn_dense_ntl.pyx b/src/sage/rings/polynomial/polynomial_modn_dense_ntl.pyx
index 04bef814741..29c75a19d30 100644
--- a/src/sage/rings/polynomial/polynomial_modn_dense_ntl.pyx
+++ b/src/sage/rings/polynomial/polynomial_modn_dense_ntl.pyx
@@ -55,9 +55,11 @@ from sage.libs.ntl.lzz_p cimport *
 from sage.libs.ntl.lzz_pX cimport *
 from sage.libs.ntl.ZZ_pX cimport *
 
+
 def make_element(parent, args):
     return parent(*args)
 
+
 zz_p_max = NTL_SP_BOUND
 
 cdef class Polynomial_dense_mod_n(Polynomial):
@@ -126,7 +128,6 @@ cdef class Polynomial_dense_mod_n(Polynomial):
             R = parent.base_ring()
             x = _dict_to_list(x, R(0))
 
-
         elif isinstance(x, ZZX):
             self._poly = x.copy()
             return
@@ -288,7 +289,6 @@ cdef class Polynomial_dense_mod_n(Polynomial):
         q, _ = self.quo_rem(right)
         return q
 
-
     def degree(self, gen=None):
         """
         Return the degree of this polynomial.
@@ -425,6 +425,7 @@ cdef class Polynomial_dense_mod_n(Polynomial):
         """
         return small_roots(self, *args, **kwds)
 
+
 def small_roots(self, X=None, beta=1.0, epsilon=None, **kwds):
     r"""
     Let `N` be the characteristic of the base ring this polynomial
@@ -623,6 +624,7 @@ def small_roots(self, X=None, beta=1.0, epsilon=None, **kwds):
     Nbeta = N**beta
     return [root for root in roots if N.gcd(ZZ(self(root))) >= Nbeta]
 
+
 cdef class Polynomial_dense_modn_ntl_zz(Polynomial_dense_mod_n):
     r"""
     Polynomial on `\ZZ/n\ZZ` implemented via NTL.
@@ -1233,7 +1235,6 @@ cdef class Polynomial_dense_modn_ntl_zz(Polynomial_dense_mod_n):
             return self._parent(int(fx))
 
 
-
 cdef class Polynomial_dense_modn_ntl_ZZ(Polynomial_dense_mod_n):
 
     def __init__(self, parent, v=None, check=True, is_gen=False, construct=False):
@@ -1260,7 +1261,6 @@ cdef class Polynomial_dense_modn_ntl_ZZ(Polynomial_dense_mod_n):
         y._parent = self._parent
         return y
 
-
     cpdef list list(self, bint copy=True):
         return [self.get_unsafe(n) for n from 0 <= n <= self.degree()]
 
@@ -1596,7 +1596,6 @@ cdef class Polynomial_dense_modn_ntl_ZZ(Polynomial_dense_mod_n):
         ZZ_pX_RightShift(r.x, self.x, n)
         return r
 
-
     def _derivative(self, var=None):
         r"""
         Return the formal derivative of ``self`` with respect to ``var``.
@@ -1636,7 +1635,6 @@ cdef class Polynomial_dense_modn_ntl_ZZ(Polynomial_dense_mod_n):
         ZZ_pX_diff(r.x, self.x)
         return r
 
-
     def reverse(self, degree=None):
         """
         Return the reverse of the input polynomial thought as a polynomial of
diff --git a/src/sage/rings/power_series_poly.pyx b/src/sage/rings/power_series_poly.pyx
index 27b1c93f7ea..cb0b0fcad1e 100644
--- a/src/sage/rings/power_series_poly.pyx
+++ b/src/sage/rings/power_series_poly.pyx
@@ -864,7 +864,6 @@ cdef class PowerSeries_poly(PowerSeries):
             except AttributeError:
                 raise ValueError('cannot differentiate with respect to {}'.format(var))
 
-
         # compute formal derivative with respect to generator
         return PowerSeries_poly(self._parent, self.__f._derivative(),
                                 self.prec()-1, check=False)
@@ -1246,6 +1245,7 @@ def make_powerseries_poly_v0(parent, f, prec, is_gen):
     """
     return PowerSeries_poly(parent, f, prec, 0, is_gen)
 
+
 cdef class BaseRingFloorDivAction(Action):
     """
     The floor division action of the base ring on a formal power series.
diff --git a/src/sage/rings/real_arb.pyx b/src/sage/rings/real_arb.pyx
index 971e9fc13bf..73232348a01 100644
--- a/src/sage/rings/real_arb.pyx
+++ b/src/sage/rings/real_arb.pyx
@@ -1148,6 +1148,7 @@ class RealBallField(UniqueRepresentation, sage.rings.abc.RealBallField):
         """
         return ARF_PREC_EXACT
 
+
 cdef inline bint _do_sig(long prec) noexcept:
     """
     Whether signal handlers should be installed for calls to FLINT.
@@ -1162,6 +1163,7 @@ cdef inline bint _do_sig(long prec) noexcept:
 cdef inline long prec(RealBall ball) noexcept:
     return ball._parent._prec
 
+
 def create_RealBall(parent, serialized):
     r"""
     Create a RealBall from a serialized representation.
@@ -1186,6 +1188,7 @@ def create_RealBall(parent, serialized):
     else:
         return res
 
+
 cdef class RealBall(RingElement):
     """
     Hold one ``arb_t``
diff --git a/src/sage/rings/real_lazy.pyx b/src/sage/rings/real_lazy.pyx
index 1ba394741fa..4b241818d84 100644
--- a/src/sage/rings/real_lazy.pyx
+++ b/src/sage/rings/real_lazy.pyx
@@ -354,6 +354,7 @@ class RealLazyField_class(LazyField):
 
 RLF = RealLazyField_class()
 
+
 def RealLazyField():
     """
     Return the lazy real field.
@@ -509,6 +510,7 @@ class ComplexLazyField_class(LazyField):
 
 CLF = ComplexLazyField_class()
 
+
 def ComplexLazyField():
     """
     Returns the lazy complex field.
@@ -523,7 +525,6 @@ def ComplexLazyField():
     return CLF
 
 
-
 cdef int get_new_prec(R, int depth) except -1:
     """
     There are depth operations, so we want at least that many more digits of
@@ -917,6 +918,7 @@ def make_element(parent, *args):
     """
     return parent(*args)
 
+
 cdef class LazyWrapper(LazyFieldElement):
 
     cpdef int depth(self) noexcept:
diff --git a/src/sage/rings/semirings/tropical_semiring.pyx b/src/sage/rings/semirings/tropical_semiring.pyx
index 2922298e286..aa3ea3da761 100644
--- a/src/sage/rings/semirings/tropical_semiring.pyx
+++ b/src/sage/rings/semirings/tropical_semiring.pyx
@@ -418,6 +418,7 @@ cdef class TropicalSemiringElement(Element):
             return infinity
         return self._val
 
+
 class TropicalSemiring(Parent, UniqueRepresentation):
     r"""
     The tropical semiring.
@@ -651,6 +652,7 @@ class TropicalSemiring(Parent, UniqueRepresentation):
         """
         return (self.element_class(self, self.base().one()), self.infinity())
 
+
 cdef class TropicalToTropical(Map):
     """
     Map from the tropical semiring to itself (possibly with different bases).