diff --git a/src/sage/rings/function_field/function_field_polymod.py b/src/sage/rings/function_field/function_field_polymod.py
index b740b771bec..44c1390bc86 100644
--- a/src/sage/rings/function_field/function_field_polymod.py
+++ b/src/sage/rings/function_field/function_field_polymod.py
@@ -418,10 +418,11 @@ def monic_integral_model(self, names=None):
                 raise ValueError("names must contain at most 2 entries")
 
         if self.base_field() is not self.rational_function_field():
-            L,from_L,to_L = self.simple_model()
-            ret,ret_to_L,L_to_ret = L.monic_integral_model(names)
-            from_ret = ret.hom( [from_L(ret_to_L(ret.gen())), from_L(ret_to_L(ret.base_field().gen()))] )
-            to_ret = self.hom( [L_to_ret(to_L(k.gen())) for k in self._intermediate_fields(self.rational_function_field())] )
+            L, from_L, to_L = self.simple_model()
+            ret, ret_to_L, L_to_ret = L.monic_integral_model(names)
+            from_ret = ret.hom([from_L(ret_to_L(ret.gen())),
+                                from_L(ret_to_L(ret.base_field().gen()))])
+            to_ret = self.hom([L_to_ret(to_L(k.gen())) for k in self._intermediate_fields(self.rational_function_field())])
             return ret, from_ret, to_ret
         else:
             if self.polynomial().is_monic() and all(c.denominator().is_one() for c in self.polynomial()):
@@ -431,12 +432,12 @@ def monic_integral_model(self, names=None):
                 return self.change_variable_name(names)
             else:
                 if not names:
-                    names = (self.variable_name()+"_",)
+                    names = (self.variable_name() + "_",)
                 if len(names) == 1:
                     names = (names[0], self.rational_function_field().variable_name())
 
                 g, d = self._make_monic_integral(self.polynomial())
-                K,from_K,to_K = self.base_field().change_variable_name(names[1])
+                K, from_K, to_K = self.base_field().change_variable_name(names[1])
                 g = g.map_coefficients(to_K)
                 ret = K.extension(g, names=names[0])
                 from_ret = ret.hom([self.gen() * d, self.base_field().gen()])
@@ -791,7 +792,7 @@ def free_module(self, base=None, basis=None, map=True):
         if not map:
             return V
         from_V = MapVectorSpaceToFunctionField(V, self)
-        to_V   = MapFunctionFieldToVectorSpace(self, V)
+        to_V = MapFunctionFieldToVectorSpace(self, V)
         return (V, from_V, to_V)
 
     def maximal_order(self):
@@ -982,7 +983,7 @@ def hom(self, im_gens, base_morphism=None):
                     yy |--> y
 
         """
-        if not isinstance(im_gens, (list,tuple)):
+        if not isinstance(im_gens, (list, tuple)):
             im_gens = [im_gens]
         if len(im_gens) == 0:
             raise ValueError("no images specified")
@@ -1025,11 +1026,11 @@ def genus(self):
             # making the auxiliary ring which only has polynomials
             # with integral coefficients.
             tmpAuxRing = PolynomialRing(self._base_field.constant_field(),
-                            str(self._base_field.gen())+','+str(self._ring.gen()))
+                            str(self._base_field.gen()) + ',' + str(self._ring.gen()))
             intMinPoly, d = self._make_monic_integral(self._polynomial)
             curveIdeal = tmpAuxRing.ideal(intMinPoly)
 
-            singular.lib('normal.lib') #loading genus method in Singular
+            singular.lib('normal.lib')  # loading genus method in Singular
             return int(curveIdeal._singular_().genus())
 
         else:
@@ -1170,7 +1171,7 @@ def _simple_model(self, name='v'):
         B = MS(B)
         M_b = V_to_N(B.solve_left(M_to_V(b)))
         M_a = V_to_N(B.solve_left(M_to_V(a)))
-        M_to_N = M.hom([M_a,M_b])
+        M_to_N = M.hom([M_a, M_b])
 
         return N, N_to_M, M_to_N
 
@@ -1285,7 +1286,7 @@ def simple_model(self, name=None):
         if isinstance(self.base_field(), RationalFunctionField):
             # the extension is simple already
             if name == self.variable_name():
-                id = Hom(self,self).identity()
+                id = Hom(self, self).identity()
                 return self, id, id
             else:
                 ret = self.base_field().extension(self.polynomial(), names=(name,))
@@ -1299,8 +1300,8 @@ def simple_model(self, name=None):
             self_ = base_.extension(self.polynomial().map_coefficients(to_base_), names=(name,))
             gens_in_base_ = [to_base_(k.gen())
                              for k in base._intermediate_fields(base.rational_function_field())]
-            to_self_ = self.hom([self_.gen()]+gens_in_base_)
-            from_self_ = self_.hom([self.gen(),from_base_(base_.gen())])
+            to_self_ = self.hom([self_.gen()] + gens_in_base_)
+            from_self_ = self_.hom([self.gen(), from_base_(base_.gen())])
 
             # now collapse self_/base_/K(x)
             ret, ret_to_self_, self__to_ret = self_._simple_model(name)
@@ -1549,8 +1550,8 @@ def separable_model(self, names=None):
             from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
             R = PolynomialRing(self.constant_base_field(), names=names)
             S = R.remove_var(names[1])
-            f = R( L.polynomial().change_variable_name(names[1]).map_coefficients(
-                     lambda c:c.numerator().change_variable_name(names[0]), S))
+            f = R(L.polynomial().change_variable_name(names[1]).map_coefficients(
+                lambda c: c.numerator().change_variable_name(names[0]), S))
             f = f.polynomial(R.gen(0)).change_ring(K)
             f /= f.leading_coefficient()
             # f must be separable in the other variable (otherwise it would factor)
@@ -1558,11 +1559,11 @@ def separable_model(self, names=None):
 
             ret = K.extension(f, names=(names[0],))
             # isomorphisms between L and ret are given by swapping generators
-            ret_to_L = ret.hom( [L(L.base_field().gen()), L.gen()] )
-            L_to_ret = L.hom( [ret(K.gen()), ret.gen()] )
+            ret_to_L = ret.hom([L(L.base_field().gen()), L.gen()])
+            L_to_ret = L.hom([ret(K.gen()), ret.gen()])
             # compose with from_L and to_L to get the desired isomorphisms between self and ret
-            f = ret.hom( [from_L(ret_to_L(ret.gen())), from_L(ret_to_L(ret.base_field().gen()))] )
-            t = self.hom( [L_to_ret(to_L(self.gen())), L_to_ret(to_L(self.base_field().gen()))] )
+            f = ret.hom([from_L(ret_to_L(ret.gen())), from_L(ret_to_L(ret.base_field().gen()))])
+            t = self.hom([L_to_ret(to_L(self.gen())), L_to_ret(to_L(self.base_field().gen()))])
             return ret, f, t
 
     def change_variable_name(self, name):
@@ -1637,13 +1638,13 @@ def change_variable_name(self, name):
             raise ValueError("name must contain at least one string")
         elif len(name) == 1:
             base = self.base_field()
-            from_base = to_base = Hom(base,base).identity()
+            from_base = to_base = Hom(base, base).identity()
         else:
             base, from_base, to_base = self.base_field().change_variable_name(name[1:])
 
         ret = base.extension(self.polynomial().map_coefficients(to_base), names=(name[0],))
-        f = ret.hom( [k.gen() for k in self._intermediate_fields(self.rational_function_field())] )
-        t = self.hom( [k.gen() for k in ret._intermediate_fields(ret.rational_function_field())] )
+        f = ret.hom([k.gen() for k in self._intermediate_fields(self.rational_function_field())])
+        t = self.hom([k.gen() for k in ret._intermediate_fields(ret.rational_function_field())])
         return ret, f, t
 
 
@@ -1701,7 +1702,7 @@ def _inversion_isomorphism(self):
                              x |--> x)
         """
         K = self.base_field()
-        R = PolynomialRing(K,'T')
+        R = PolynomialRing(K, 'T')
         x = K.gen()
         xinv = 1/x
 
@@ -1709,8 +1710,8 @@ def _inversion_isomorphism(self):
         F_poly = R([h(c) for c in self.polynomial().list()])
         F = K.extension(F_poly)
 
-        self2F = self.hom([F.gen(),xinv])
-        F2self = F.hom([self.gen(),xinv])
+        self2F = self.hom([F.gen(), xinv])
+        F2self = F.hom([self.gen(), xinv])
 
         M, M2F, F2M = F.monic_integral_model('s')
 
@@ -1844,7 +1845,7 @@ def genus(self):
         The genus is computed by the Hurwitz genus formula.
         """
         k, _ = self.exact_constant_field()
-        different_degree = self.different().degree() # must be even
+        different_degree = self.different().degree()  # must be even
         return Integer(different_degree // 2 - self.degree() / k.degree()) + 1
 
     def residue_field(self, place, name=None):
@@ -2123,7 +2124,7 @@ def _places_finite(self, degree):
 
         for d in degree.divisors():
             for p in K._places_finite(degree=d):
-                for prime,_,_ in O.decomposition(p.prime_ideal()):
+                for prime, _, _ in O.decomposition(p.prime_ideal()):
                     place = prime.place()
                     if place.degree() == degree:
                         yield place
@@ -2167,7 +2168,7 @@ def _places_infinite(self, degree):
             <generator object ...>
         """
         Oinf = self.maximal_order_infinite()
-        for prime,_,_ in Oinf.decomposition():
+        for prime, _, _ in Oinf.decomposition():
             place = prime.place()
             if place.degree() == degree:
                 yield place
@@ -2312,7 +2313,7 @@ def number_of_rational_places(self, r=1):
         L = self.L_polynomial()
         Lp = L.derivative()
 
-        R = IntegerRing()[[L.parent().gen()]] # power series ring
+        R = IntegerRing()[[L.parent().gen()]]  # power series ring
 
         f = R(Lp / L, prec=r)
         n = f[r-1] + q**r + 1
@@ -2408,13 +2409,13 @@ def _maximal_order_basis(self):
         from .hermite_form_polynomial import reversed_hermite_form
 
         k = self.constant_base_field()
-        K = self.base_field() # rational function field
+        K = self.base_field()  # rational function field
         n = self.degree()
 
         # Construct the defining polynomial of the function field as a
         # two-variate polynomial g in the ring k[y,x] where k is the constant
         # base field.
-        S,(y,x) = PolynomialRing(k, names='y,x', order='lex').objgens()
+        S, (y, x) = PolynomialRing(k, names='y,x', order='lex').objgens()
         v = self.polynomial().list()
         g = sum([v[i].numerator().subs(x) * y**i for i in range(len(v))])
 
@@ -2431,7 +2432,7 @@ def _maximal_order_basis(self):
             gflat = R.zero()
             for m in g.monomials():
                 c = g.monomial_coefficient(m).polynomial('zz')
-                gflat += R(c) * R(m) # R(m) is a monomial in yy and xx
+                gflat += R(c) * R(m)  # R(m) is a monomial in yy and xx
 
             k_poly = R(k.polynomial('zz'))
 
@@ -2439,7 +2440,7 @@ def _maximal_order_basis(self):
             pols_in_R = normalize(R.ideal([k_poly, gflat]))
 
             # reconstruct polynomials in S
-            h = R.hom([y,x,k.gen()],S)
+            h = R.hom([y, x, k.gen()], S)
             pols_in_S = [h(f) for f in pols_in_R]
         else:
             # Call Singular. Singular's "normal" function returns a basis
diff --git a/src/sage/rings/function_field/function_field_rational.py b/src/sage/rings/function_field/function_field_rational.py
index 226d87d084c..8a0a6f25764 100644
--- a/src/sage/rings/function_field/function_field_rational.py
+++ b/src/sage/rings/function_field/function_field_rational.py
@@ -323,7 +323,7 @@ def _to_polynomial(self, f):
         """
         K = f.parent().constant_base_field()
         if f.denominator() in K:
-            return f.numerator()/K(f.denominator())
+            return f.numerator() / K(f.denominator())
         raise ValueError("only polynomials can be converted to the underlying polynomial ring")
 
     def _to_bivariate_polynomial(self, f):
@@ -350,7 +350,8 @@ def _to_bivariate_polynomial(self, f):
         v = f.list()
         denom = lcm([a.denominator() for a in v])
         S = denom.parent()
-        x,t = S.base_ring()['%s,%s' % (f.parent().variable_name(),self.variable_name())].gens()
+        x, t = S.base_ring()['%s,%s' % (f.parent().variable_name(),
+                                        self.variable_name())].gens()
         phi = S.hom([t])
         return sum([phi((denom * v[i]).numerator()) * x**i for i in range(len(v))]), denom
 
@@ -420,12 +421,12 @@ def _factor_univariate_polynomial(self, f, proof=None):
         x = f.parent().gen()
         t = f.parent().base_ring().gen()
         phi = F.parent().hom([x, t])
-        v = [(phi(P),e) for P, e in fac]
-        unit = phi(fac.unit())/d
+        v = [(phi(P), e) for P, e in fac]
+        unit = phi(fac.unit()) / d
         w = []
         for a, e in v:
             c = a.leading_coefficient()
-            a = a/c
+            a = a / c
             # undo any variable substitution that we introduced for the bivariate polynomial
             if old_variable_name != a.variable_name():
                 a = a.change_variable_name(old_variable_name)
@@ -433,7 +434,7 @@ def _factor_univariate_polynomial(self, f, proof=None):
             if a.is_unit():
                 unit *= a**e
             else:
-                w.append((a,e))
+                w.append((a, e))
         from sage.structure.factorization import Factorization
         return Factorization(w, unit=unit)
 
@@ -553,7 +554,7 @@ def free_module(self, base=None, basis=None, map=True):
         if not map:
             return V
         from_V = MapVectorSpaceToFunctionField(V, self)
-        to_V   = MapFunctionFieldToVectorSpace(self, V)
+        to_V = MapFunctionFieldToVectorSpace(self, V)
         return (V, from_V, to_V)
 
     def random_element(self, *args, **kwds):
@@ -683,7 +684,7 @@ def hom(self, im_gens, base_morphism=None):
         """
         if isinstance(im_gens, CategoryObject):
             return self.Hom(im_gens).natural_map()
-        if not isinstance(im_gens, (list,tuple)):
+        if not isinstance(im_gens, (list, tuple)):
             im_gens = [im_gens]
         if len(im_gens) != 1:
             raise ValueError("there must be exactly one generator")
@@ -834,8 +835,8 @@ def change_variable_name(self, name):
                 raise ValueError("names must be a tuple with a single string")
             name = name[0]
         if name == self.variable_name():
-            id = Hom(self,self).identity()
-            return self,id,id
+            id = Hom(self, self).identity()
+            return self, id, id
         else:
             from .constructor import FunctionField
             ret = FunctionField(self.constant_base_field(), name)
diff --git a/src/sage/rings/function_field/ideal_polymod.py b/src/sage/rings/function_field/ideal_polymod.py
index 6735dd6a8fa..da45bc508cf 100644
--- a/src/sage/rings/function_field/ideal_polymod.py
+++ b/src/sage/rings/function_field/ideal_polymod.py
@@ -27,6 +27,7 @@
 
 from .ideal import FunctionFieldIdeal, FunctionFieldIdealInfinite
 
+
 class FunctionFieldIdeal_polymod(FunctionFieldIdeal):
     """
     Fractional ideals of algebraic function fields
@@ -407,7 +408,7 @@ def _mul_(self, other):
                 g1, g2 = other._gens_two_vecs
                 vecs = list(g1 * self._hnf) + [mul(g2, v) for v in self._hnf]
         else:
-            vecs = [mul(r1,r2) for r1 in self._hnf for r2 in other._hnf]
+            vecs = [mul(r1, r2) for r1 in self._hnf for r2 in other._hnf]
 
         return O._ideal_from_vectors_and_denominator(vecs, self._denominator * other._denominator)
 
@@ -488,7 +489,7 @@ def intersect(self, other):
         n = A.ncols()
 
         # intersect the row spaces of A and B
-        M = block_matrix([[I,I],[A,O],[O,B]])
+        M = block_matrix([[I, I], [A, O], [O, B]])
 
         # reversed Hermite form
         U = reversed_hermite_form(M, transformation=True)
@@ -601,7 +602,7 @@ def gens_over_base(self):
             sage: I.gens_over_base()
             (x^3 + 1, y + x)
         """
-        gens, d  = self._gens_over_base
+        gens, d = self._gens_over_base
         return tuple([~d * b for b in gens])
 
     @lazy_attribute
@@ -620,9 +621,8 @@ def _gens_over_base(self):
             sage: I._gens_over_base
             ([x, y], x)
         """
-        gens = []
-        for row in self._hnf:
-            gens.append(sum([c1*c2 for c1,c2 in zip(row, self._ring.basis())]))
+        gens = [sum([c1 * c2 for c1, c2 in zip(row, self._ring.basis())])
+                for row in self._hnf]
         return gens, self._denominator
 
     def gens(self):
@@ -864,7 +864,7 @@ def is_prime(self):
             [True, True]
         """
         factors = self.factor()
-        if len(factors) == 1 and factors[0][1] == 1: # prime!
+        if len(factors) == 1 and factors[0][1] == 1:  # prime!
             prime = factors[0][0]
             assert self == prime
             self._relative_degree = prime._relative_degree
@@ -1013,13 +1013,13 @@ def _factor(self):
         i = d * self
 
         factors = []
-        primes = set([o.ideal(p) for p,_ in d.factor()] + [p for p,_ in i.ideal_below().factor()])
+        primes = set([o.ideal(p) for p, _ in d.factor()] + [p for p, _ in i.ideal_below().factor()])
         for prime in primes:
             qs = [q[0] for q in O.decomposition(prime)]
             for q in qs:
                 exp = q.valuation(self)
                 if exp != 0:
-                    factors.append((q,exp))
+                    factors.append((q, exp))
         return factors
 
 
@@ -1100,10 +1100,10 @@ def __pow__(self, mod):
                 J = [ppow * v for v in I]
             else:
                 p, q = self._gens_two_vecs
-                q = sum(e1 * e2 for e1,e2 in zip(O.basis(), q))
+                q = sum(e1 * e2 for e1, e2 in zip(O.basis(), q))
                 ppow = p**mod
                 qpow = O._coordinate_vector(q**mod)
-                J = [ppow * v for v in I] + [mul(qpow,v) for v in I]
+                J = [ppow * v for v in I] + [mul(qpow, v) for v in I]
 
             return O._ideal_from_vectors_and_denominator(J, self._denominator**mod)
 
@@ -1171,7 +1171,7 @@ def gens_two(self):
             of Function field in y defined by y^2 + y + (x^2 + 1)/x
         """
         d = self.denominator()
-        return tuple(e/d for e in self._gens_two())
+        return tuple(e / d for e in self._gens_two())
 
     @cached_method
     def _gens_two(self):
@@ -1219,18 +1219,18 @@ def _gens_two(self):
         O = self.ring()
         F = O.fraction_field()
 
-        if self._kummer_form is not None: # prime ideal
+        if self._kummer_form is not None:  # prime ideal
             _g1, _g2 = self._kummer_form
             g1 = F(_g1)
-            g2 = sum([c1*c2 for c1,c2 in zip(_g2, O.basis())])
+            g2 = sum([c1 * c2 for c1, c2 in zip(_g2, O.basis())])
             if g2:
                 self._gens_two_vecs = (_g1, _g2)
-                return (g1,g2)
+                return (g1, g2)
             else:
                 self._gens_two_vecs = (_g1,)
                 return (g1,)
 
-        ### start to search for two generators
+        # ---- start to search for two generators
 
         hnf = self._hnf
 
@@ -1238,7 +1238,7 @@ def _gens_two(self):
         for e in hnf.diagonal():
             norm *= e
 
-        if norm.is_constant(): # unit ideal
+        if norm.is_constant():  # unit ideal
             self._gens_two_vecs = (1,)
             return (F(1),)
 
@@ -1247,7 +1247,7 @@ def _gens_two(self):
         p = _l.degree()
         l = F(_l)
 
-        if self._hnf == O.ideal(l)._hnf: # principal ideal
+        if self._hnf == O.ideal(l)._hnf:  # principal ideal
             self._gens_two_vecs = (_l,)
             return (l,)
 
@@ -1255,13 +1255,13 @@ def _gens_two(self):
 
         basis = []
         for row in hnf:
-            basis.append(sum([c1*c2 for c1,c2 in zip(row, O.basis())]))
+            basis.append(sum([c1 * c2 for c1, c2 in zip(row, O.basis())]))
 
         n = len(basis)
         alpha = None
 
         def check(alpha):
-            alpha_norm = alpha.norm().numerator() # denominator is 1
+            alpha_norm = alpha.norm().numerator()  # denominator is 1
             return norm.gcd(alpha_norm // norm) == 1
 
         # Trial 1: search for alpha among generators
@@ -1273,9 +1273,9 @@ def check(alpha):
         # Trial 2: exhaustive search for alpha using only polynomials
         # with coefficients 0 or 1
         for d in range(p):
-            G = itertools.product(itertools.product([0,1],repeat=d+1), repeat=n)
+            G = itertools.product(itertools.product([0, 1], repeat=d + 1), repeat=n)
             for g in G:
-                alpha = sum([R(c1)*c2 for c1,c2 in zip(g, basis)])
+                alpha = sum([R(c1) * c2 for c1, c2 in zip(g, basis)])
                 if check(alpha):
                     self._gens_two_vecs = (_l, O._coordinate_vector(alpha))
                     return (l, alpha)
@@ -1293,7 +1293,7 @@ def check(alpha):
                 if g[j].leading_coefficient() != 1:
                     continue
 
-                alpha = sum([c1*c2 for c1,c2 in zip(g, basis)])
+                alpha = sum([c1 * c2 for c1, c2 in zip(g, basis)])
                 if check(alpha):
                     self._gens_two_vecs = (_l, O._coordinate_vector(alpha))
                     return (l, alpha)
@@ -1383,7 +1383,7 @@ def __contains__(self, x):
             True
         """
         F = self.ring().fraction_field()
-        iF,from_iF,to_iF = F._inversion_isomorphism()
+        iF, from_iF, to_iF = F._inversion_isomorphism()
         return to_iF(x) in self._ideal
 
     def _add_(self, other):
@@ -1580,7 +1580,7 @@ def gens(self):
             (x, y)
         """
         F = self.ring().fraction_field()
-        iF,from_iF,to_iF = F._inversion_isomorphism()
+        iF, from_iF, to_iF = F._inversion_isomorphism()
         return tuple(from_iF(b) for b in self._ideal.gens())
 
     def gens_two(self):
@@ -1606,7 +1606,7 @@ def gens_two(self):
             (x,)
         """
         F = self.ring().fraction_field()
-        iF,from_iF,to_iF = F._inversion_isomorphism()
+        iF, from_iF, to_iF = F._inversion_isomorphism()
         return tuple(from_iF(b) for b in self._ideal.gens_two())
 
     def gens_over_base(self):
@@ -1624,7 +1624,7 @@ def gens_over_base(self):
             (x, y, 1/x^2*y^2)
         """
         F = self.ring().fraction_field()
-        iF,from_iF,to_iF = F._inversion_isomorphism()
+        iF, from_iF, to_iF = F._inversion_isomorphism()
         return tuple(from_iF(g) for g in self._ideal.gens_over_base())
 
     def ideal_below(self):
diff --git a/src/sage/rings/function_field/order_polymod.py b/src/sage/rings/function_field/order_polymod.py
index 97f5a625819..067ca2ed090 100644
--- a/src/sage/rings/function_field/order_polymod.py
+++ b/src/sage/rings/function_field/order_polymod.py
@@ -547,7 +547,7 @@ def codifferent(self):
             + 6*x/(x^4 + 4*x^3 + 3*x^2 + 6*x + 4)) of Maximal order of Function field
             in y defined by y^4 + x*y + 4*x + 1
         """
-        T  = self._codifferent_matrix()
+        T = self._codifferent_matrix()
         return self._ideal_from_vectors(T.inverse().columns())
 
     @cached_method
@@ -686,7 +686,7 @@ def decomposition(self, ideal):
                 m = []
                 for g in q.basis_matrix():
                     m.extend(matrix([g * mr for mr in matrices_reduced]).columns())
-                beta  = [c.lift() for c in matrix(m).right_kernel().basis()[0]]
+                beta = [c.lift() for c in matrix(m).right_kernel().basis()[0]]
 
                 r = q
                 index = 1
@@ -1291,7 +1291,7 @@ def decomposition(self, ideal):
                 m = []
                 for i in range(n):
                     m.append(sum(qgenb[j] * mtable[i][j] for j in range(n)))
-                beta  = [fr(coeff) for coeff in matrix(m).left_kernel().basis()[0]]
+                beta = [fr(coeff) for coeff in matrix(m).left_kernel().basis()[0]]
 
                 prime.is_prime.set_cache(True)
                 prime._prime_below = ideal
diff --git a/src/sage/rings/function_field/place_polymod.py b/src/sage/rings/function_field/place_polymod.py
index e11a6c86ab7..f345c31129e 100644
--- a/src/sage/rings/function_field/place_polymod.py
+++ b/src/sage/rings/function_field/place_polymod.py
@@ -463,7 +463,7 @@ def _residue_field(self, name=None):
         prime = self.prime_ideal()  # Let P be this prime ideal
 
         if self.is_infinite_place():
-            _F, from_F, to_F  = F._inversion_isomorphism()
+            _F, from_F, to_F = F._inversion_isomorphism()
             _prime = prime._ideal
             _place = _prime.place()
 
diff --git a/src/sage/rings/number_field/number_field_ideal.py b/src/sage/rings/number_field/number_field_ideal.py
index 8a7f6272da5..b80a4a407ef 100644
--- a/src/sage/rings/number_field/number_field_ideal.py
+++ b/src/sage/rings/number_field/number_field_ideal.py
@@ -3503,10 +3503,10 @@ def quotient_char_p(I, p):
     if not I.is_integral():
         raise ValueError("I must be an integral ideal.")
 
-    K    = I.number_field()
-    OK   = K.maximal_order()  # will in the long run only really need a p-maximal order.
+    K = I.number_field()
+    OK = K.maximal_order()  # will in the long run only really need a p-maximal order.
     M_OK = OK.free_module()
-    M_I  = I.free_module()
+    M_I = I.free_module()
 
     # Now we have to quite explicitly find a way to compute
     # with OK / I viewed as a quotient of two F_p vector spaces,
diff --git a/src/sage/rings/number_field/selmer_group.py b/src/sage/rings/number_field/selmer_group.py
index 37abe43101e..87221473348 100644
--- a/src/sage/rings/number_field/selmer_group.py
+++ b/src/sage/rings/number_field/selmer_group.py
@@ -129,7 +129,7 @@ def _coords_in_C_p(I, C, p):
     """
     cyclic_orders = C.gens_orders()
     non_p_indices = [i for i,n in enumerate(cyclic_orders) if not p.divides(n)]
-    p_indices     = [(i, n // p) for i,n in enumerate(cyclic_orders) if p.divides(n)]
+    p_indices = [(i, n // p) for i,n in enumerate(cyclic_orders) if p.divides(n)]
 
     coords = C(I).exponents()
     if all(coords[i] == 0 for i in non_p_indices) and all(coords[i] % n == 0 for i, n in p_indices):
@@ -227,7 +227,7 @@ def _root_ideal(I, C, p):
 
     """
     cyclic_orders = C.gens_orders()
-    cyclic_gens   = C.gens_ideals()
+    cyclic_gens = C.gens_ideals()
     coords = C(I).exponents()
 
     # In the next line, e=(ci/p)%n should satisfy p*e=ci (mod n): we
@@ -651,7 +651,7 @@ def to_KSp(a):
         if debug:
             assert all(v % p == 0 for v in vals)
         one = K(1) if K == QQ else K.ideal(1)
-        aa  = a.abs() if K == QQ else K.ideal(a)
+        aa = a.abs() if K == QQ else K.ideal(a)
         B = prod((P ** (v // p) for P, v in zip(supp,vals)), one)
         if debug:
             assert B ** p == aa
diff --git a/src/sage/rings/polynomial/multi_polynomial_ideal.py b/src/sage/rings/polynomial/multi_polynomial_ideal.py
index fa6655ca516..8f10f30add7 100644
--- a/src/sage/rings/polynomial/multi_polynomial_ideal.py
+++ b/src/sage/rings/polynomial/multi_polynomial_ideal.py
@@ -405,7 +405,7 @@ def _groebner_basis_magma(self, deg_bound=None, prot=False, magma=magma_default)
             sage: len(gb)
             5
         """
-        R   = self.ring()
+        R = self.ring()
         if not deg_bound:
             mself = magma(self)
         else:
@@ -3615,7 +3615,7 @@ def std(self):
 
         ALGORITHM: Uses Singular's ``std`` command
         """
-        if self.side()  == 'twosided':
+        if self.side() == 'twosided':
             return self.twostd()
         return self.ring().ideal( self.__call_singular('std'), side=self.side())
 #        return self.__call_singular('std')
diff --git a/src/sage/rings/polynomial/pbori/ll.py b/src/sage/rings/polynomial/pbori/ll.py
index a48e7e2a84a..71a01bd3b50 100644
--- a/src/sage/rings/polynomial/pbori/ll.py
+++ b/src/sage/rings/polynomial/pbori/ll.py
@@ -140,7 +140,7 @@ def eliminate_ll_ranked(ll_system, to_reduce,
                         reduction_function=ll_red_nf_noredsb,
                         reduce_ll_system=False, prot=False):
 
-    assert(ll_system)
+    assert ll_system
     from_ring = ll_system[0].ring()
 
     ll_ranks = rank(ll_system)
diff --git a/src/sage/rings/polynomial/term_order.py b/src/sage/rings/polynomial/term_order.py
index 687545c0c39..48e49ac89fa 100644
--- a/src/sage/rings/polynomial/term_order.py
+++ b/src/sage/rings/polynomial/term_order.py
@@ -802,7 +802,7 @@ def __init__(self, name='lex', n=0, force=False):
                     singular_str = []
                     macaulay2_str = []
 
-                    length_pattern  = re.compile(r"\(([0-9]+)\)$") # match with parenthesized block length at end
+                    length_pattern = re.compile(r"\(([0-9]+)\)$") # match with parenthesized block length at end
                     for block in block_names:
                         try:
                             block_name, block_length, _ = re.split(length_pattern,block.strip())
@@ -1312,7 +1312,7 @@ def greater_tuple_deglex(self,f,g):
         """
         sf = sum(f.nonzero_values(sort=False))
         sg = sum(g.nonzero_values(sort=False))
-        return ( sf > sg or ( sf == sg and f  > g )) and f or g
+        return ( sf > sg or ( sf == sg and f > g )) and f or g
 
     def greater_tuple_degrevlex(self,f,g):
         """
@@ -1396,7 +1396,7 @@ def greater_tuple_negdeglex(self,f,g):
         """
         sf = sum(f.nonzero_values(sort=False))
         sg = sum(g.nonzero_values(sort=False))
-        return ( sf < sg or ( sf == sg and f  > g )) and f or g
+        return ( sf < sg or ( sf == sg and f > g )) and f or g
 
     def greater_tuple_degneglex(self,f,g):
         """
@@ -1474,7 +1474,7 @@ def greater_tuple_wdeglex(self,f,g):
         """
         sf = sum(l*r for (l,r) in zip(f,self._weights))
         sg = sum(l*r for (l,r) in zip(g,self._weights))
-        return (sf > sg or ( sf == sg and f  > g )) and f or g
+        return (sf > sg or ( sf == sg and f > g )) and f or g
 
     def greater_tuple_wdegrevlex(self,f,g):
         """
@@ -1531,7 +1531,7 @@ def greater_tuple_negwdeglex(self,f,g):
         """
         sf = sum(l*r for (l,r) in zip(f,self._weights))
         sg = sum(l*r for (l,r) in zip(g,self._weights))
-        return (sf < sg or ( sf == sg and f  > g )) and f or g
+        return (sf < sg or ( sf == sg and f > g )) and f or g
 
     def greater_tuple_negwdegrevlex(self,f,g):
         """
diff --git a/src/sage/rings/quotient_ring_element.py b/src/sage/rings/quotient_ring_element.py
index f67c7192f11..56829641624 100644
--- a/src/sage/rings/quotient_ring_element.py
+++ b/src/sage/rings/quotient_ring_element.py
@@ -422,7 +422,7 @@ def _div_(self, right):
 
         # We are computing L/R modulo the ideal.
         (L, R) = (self.__rep, right.__rep)
-        P  = self.parent()
+        P = self.parent()
         I = P.defining_ideal()
 
         if not hasattr(I, 'groebner_basis'):
diff --git a/src/sage_docbuild/conf.py b/src/sage_docbuild/conf.py
index 6b84e772dd1..f940adb0755 100644
--- a/src/sage_docbuild/conf.py
+++ b/src/sage_docbuild/conf.py
@@ -57,7 +57,7 @@
 
 jupyter_execute_default_kernel = 'sagemath'
 
-if os.environ.get('SAGE_LIVE_DOC', 'no')  == 'yes':
+if os.environ.get('SAGE_LIVE_DOC', 'no') == 'yes':
     SAGE_JUPYTER_SERVER = os.environ.get('SAGE_JUPYTER_SERVER', 'binder')
     if SAGE_JUPYTER_SERVER.startswith('binder'):
         # format: "binder" or
@@ -197,7 +197,7 @@ def sphinx_plot(graphics, **kwds):
 
 # The language for content autogenerated by Sphinx. Refer to documentation
 # for a list of supported languages.
-#language = None
+# language = None
 
 # The LaTeX engine to build the docs.
 # https://www.sphinx-doc.org/en/master/usage/configuration.html#confval-latex_engine
@@ -205,9 +205,9 @@ def sphinx_plot(graphics, **kwds):
 
 # There are two options for replacing |today|: either, you set today to some
 # non-false value, then it is used:
-#today = ''
+# today = ''
 # Else, today_fmt is used as the format for a strftime call.
-#today_fmt = '%B %d, %Y'
+# today_fmt = '%B %d, %Y'
 
 # List of glob-style patterns that should be excluded when looking for
 # source files. [1] They are matched against the source file names
@@ -216,11 +216,11 @@ def sphinx_plot(graphics, **kwds):
 exclude_patterns = ['.build']
 
 # If true, '()' will be appended to :func: etc. cross-reference text.
-#add_function_parentheses = True
+# add_function_parentheses = True
 
 # If true, the current module name will be prepended to all description
 # unit titles (such as .. function::).
-#add_module_names = True
+# add_module_names = True
 
 # If true, sectionauthor and moduleauthor directives will be shown in the
 # output. They are ignored by default.
@@ -274,9 +274,9 @@ def set_intersphinx_mappings(app, config):
         return
 
     app.config.intersphinx_mapping = {
-    'python': ('https://docs.python.org/',
-                os.path.join(SAGE_DOC_SRC, "common",
-                             "python{}.inv".format(python_version))),
+        'python': ('https://docs.python.org/',
+                   os.path.join(SAGE_DOC_SRC, "common",
+                                "python{}.inv".format(python_version))),
     }
     if PPLPY_DOCS and os.path.exists(os.path.join(PPLPY_DOCS, 'objects.inv')):
         app.config.intersphinx_mapping['pplpy'] = (PPLPY_DOCS, None)
@@ -379,18 +379,18 @@ def set_intersphinx_mappings(app, config):
     html_theme_options = {}
 
 # HTML style sheet. This overrides a HTML theme's corresponding setting.
-#html_style = 'default.css'
+# html_style = 'default.css'
 
 # The name for this set of Sphinx documents.  If None, it defaults to
 # "<project> v<release> documentation".
-#html_title = None
+# html_title = None
 
 # A shorter title for the navigation bar.  Default is the same as html_title.
-#html_short_title = None
+# html_short_title = None
 
 # The name of an image file (within the static path) to place at the top of
 # the sidebar.
-#html_logo = 'sagelogo-word.ico'
+# html_logo = 'sagelogo-word.ico'
 
 # The name of an image file (within the static path) to use as favicon of the
 # docs.  This file should be a Windows icon file (.ico) being 16x16 or 32x32
@@ -439,17 +439,17 @@ def set_intersphinx_mappings(app, config):
 
 # If true, SmartyPants will be used to convert quotes and dashes to
 # typographically correct entities.
-#html_use_smartypants = True
+# html_use_smartypants = True
 
 # Custom sidebar templates, maps document names to template names.
-#html_sidebars = {}
+# html_sidebars = {}
 
 # Additional templates that should be rendered to pages, maps page names to
 # template names.
-#html_additional_pages = {}
+# html_additional_pages = {}
 
 # If false, no module index is generated.
-#html_use_modindex = True
+# html_use_modindex = True
 
 # A list of prefixes that are ignored for sorting the Python module index ( if
 # this is set to ['foo.'], then foo.bar is shown under B, not F). Works only
@@ -852,6 +852,7 @@ def apply(self):
 
                     node.parent.insert(node.parent.index(node) + 1, cell_node)
 
+
 # This replaces the setup() in sage.misc.sagedoc_conf
 def setup(app):
     app.connect('autodoc-process-docstring', process_docstring_cython)
diff --git a/src/tox.ini b/src/tox.ini
index f2f1fc158fd..8e562cced8e 100644
--- a/src/tox.ini
+++ b/src/tox.ini
@@ -162,7 +162,7 @@ description =
     # W605: invalid escape sequence ‘x’
     # See https://pycodestyle.pycqa.org/en/latest/intro.html#error-codes
 deps = pycodestyle
-commands = pycodestyle --select E111,E21,E222,E225,E227,E228,E25,E271,E303,E305,E306,E401,E502,E701,E702,E703,E71,E72,W291,W293,W391,W605 {posargs:{toxinidir}/sage/}
+commands = pycodestyle --select E111,E21,E221,E222,E225,E227,E228,E25,E271,E303,E305,E306,E401,E502,E701,E702,E703,E71,E72,W291,W293,W391,W605 {posargs:{toxinidir}/sage/}
        pycodestyle --select E111,E271,E301,E306,E401,E502,E703,E712,E713,E714,E72,W29,W391,W605, --filename *.pyx {posargs:{toxinidir}/sage/}
 
 [pycodestyle]