gh-38263: partial care for empty lines in pyx in rings ; activate E30…

…2 in pyx

    
this is fixing blank lines issues in pyx files in `rings` folder ; and
activating the check for pycodestyle  E302 in all pyx files

### 📝 Checklist

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- [x] The title is concise and informative.
- [x] The description explains in detail what this PR is about.
    
URL: #38263
Reported by: Frédéric Chapoton
Reviewer(s): David Coudert

src/sage/rings/complex_interval.pyx

@@ -1002,7 +1002,6 @@ cdef class ComplexIntervalFieldElement(FieldElement):
         """
         raise TypeError
 
-
     def _sage_input_(self, sib, coerce):
         r"""
         Produce an expression which will reproduce this value when evaluated.

src/sage/rings/finite_rings/element_base.pyx

@@ -24,6 +24,7 @@ from sage.rings.integer_ring import ZZ
 from sage.rings.integer import Integer
 from sage.misc.superseded import deprecated_function_alias
 
+
 def is_FiniteFieldElement(x):
     """
     Return ``True`` if ``x`` is a finite field element.

src/sage/rings/finite_rings/element_givaro.pyx

@@ -1729,6 +1729,7 @@ cdef class FiniteField_givaroElement(FinitePolyExtElement):
         """
         return unpickle_FiniteField_givaroElement,(self.parent(),self.element)
 
+
 def unpickle_FiniteField_givaroElement(parent, int x):
     """
     TESTS::

src/sage/rings/finite_rings/element_ntl_gf2e.pyx

@@ -1021,7 +1021,6 @@ cdef class FiniteField_ntl_gf2eElement(FinitePolyExtElement):
         cdef int pow = n/d
         return f if pow == 1 else f**pow
 
-
     def minpoly(self, var='x'):
         r"""
         Return the minimal polynomial of ``self``, which is the smallest
@@ -1302,6 +1301,7 @@ cdef class FiniteField_ntl_gf2eElement(FinitePolyExtElement):
         x = pari.fflog(self, base, (base_order, fac))
         return Integer(x)
 
+
 def unpickleFiniteField_ntl_gf2eElement(parent, elem):
     """
     EXAMPLES::

src/sage/rings/finite_rings/hom_finite_field_givaro.pyx

@@ -99,7 +99,6 @@ cdef class SectionFiniteFieldHomomorphism_givaro(SectionFiniteFieldHomomorphism_
 
         self._codomain_cache = (<FiniteField_givaroElement>(self._codomain.gen()))._cache
 
-
     cpdef Element _call_(self, x):
         """
         TESTS::
@@ -181,7 +180,6 @@ cdef class FiniteFieldHomomorphism_givaro(FiniteFieldHomomorphism_generic):
         self._order_domain = domain.cardinality() - 1
         self._order_codomain = codomain.cardinality() - 1
 
-
     cpdef Element _call_(self, x):
         """
         TESTS::
@@ -228,7 +226,6 @@ cdef class FrobeniusEndomorphism_givaro(FrobeniusEndomorphism_finite_field):
             raise TypeError("The domain is not an instance of FiniteField_givaro")
         FrobeniusEndomorphism_finite_field.__init__(self, domain, power)
 
-
     def fixed_field(self):
         """
         Return the fixed field of ``self``.

src/sage/rings/fraction_field_element.pyx

@@ -264,7 +264,6 @@ cdef class FractionFieldElement(FieldElement):
         """
         return self._denominator
 
-
     def is_square(self,root=False):
         """
         Return whether or not ``self`` is a perfect square.
@@ -1286,6 +1285,7 @@ cdef class FractionFieldElement_1poly_field(FractionFieldElement):
         super(self.__class__, self).reduce()
         self.normalize_leading_coefficients()
 
+
 def make_element(parent, numerator, denominator):
     """
     Used for unpickling :class:`FractionFieldElement` objects (and subclasses).

src/sage/rings/laurent_series_ring_element.pyx

@@ -149,7 +149,6 @@ cdef class LaurentSeries(AlgebraElement):
         elif parent is not f.parent():
             f = parent._power_series_ring(f)
 
-
         # self is that t^n * u:
         if not f:
             if n is infinity:
@@ -822,7 +821,6 @@ cdef class LaurentSeries(AlgebraElement):
         # 3. Subtract
         return type(self)(self._parent, f1 - f2, m)
 
-
     def add_bigoh(self, prec):
         """
         Return the truncated series at chosen precision ``prec``.
@@ -1564,7 +1562,6 @@ cdef class LaurentSeries(AlgebraElement):
         """
         return multi_derivative(self, args)
 
-
     def _derivative(self, var=None):
         """
         The formal derivative of this Laurent series with respect to var.
@@ -1624,7 +1621,6 @@ cdef class LaurentSeries(AlgebraElement):
         u = self._parent._power_series_ring(v, self.__u.prec())
         return type(self)(self._parent, u, n-1)
 
-
     def integral(self):
         r"""
         The formal integral of this Laurent series with 0 constant term.
@@ -1682,7 +1678,6 @@ cdef class LaurentSeries(AlgebraElement):
             raise ArithmeticError("Coefficients of integral cannot be coerced into the base ring")
         return type(self)(self._parent, u, n+1)
 
-
     def nth_root(self, long n, prec=None):
         r"""
         Return the ``n``-th root of this Laurent power series.

src/sage/rings/polynomial/cyclotomic.pyx

@@ -40,6 +40,7 @@ try:
 except ImportError:
     pass
 
+
 def cyclotomic_coeffs(nn, sparse=None):
     """
     Return the coefficients of the `n`-th cyclotomic polynomial

src/sage/rings/polynomial/hilbert.pyx

@@ -421,6 +421,7 @@ cdef make_children(Node D, tuple w):
      #    It may be a good idea to form the product of some of the most
      #    frequent variables. But this isn't implemented yet. TODO?
 
+
 def first_hilbert_series(I, grading=None, return_grading=False):
     """
     Return the first Hilbert series of the given monomial ideal.
@@ -550,6 +551,7 @@ def first_hilbert_series(I, grading=None, return_grading=False):
                 fmpz_poly_add(fhs._poly, AN.LMult, AN.RMult)
                 got_result = True
 
+
 def hilbert_poincare_series(I, grading=None):
     r"""
     Return the Hilbert Poincaré series of the given monomial ideal.

src/sage/rings/polynomial/multi_polynomial_ideal_libsingular.pyx

@@ -145,6 +145,7 @@ cdef ideal *sage_ideal_to_singular_ideal(I) except NULL:
             raise TypeError("All generators must be of type MPolynomial_libsingular.")
     return i
 
+
 def kbase_libsingular(I, degree=None):
     """
     SINGULAR's ``kbase()`` algorithm.
@@ -201,6 +202,7 @@ def kbase_libsingular(I, degree=None):
 
     return res
 
+
 def std_libsingular(I):
     """
     SINGULAR's ``std()`` algorithm.
@@ -224,7 +226,6 @@ def std_libsingular(I):
 
     idSkipZeroes(result)
 
-
     id_Delete(&i,r)
 
     res = singular_ideal_to_sage_sequence(result,r,I.ring())
@@ -272,6 +273,7 @@ def slimgb_libsingular(I):
     id_Delete(&result,r)
     return res
 
+
 def interred_libsingular(I):
     """
     SINGULAR's ``interred()`` command.
@@ -318,7 +320,6 @@ def interred_libsingular(I):
     sig_off()
     singular_options = bck
 
-
     # divide head by coefficients
     if r.cf.type != n_Z and r.cf.type != n_Znm and r.cf.type != n_Zn and r.cf.type != n_Z2m :
         for j from 0 <= j < IDELEMS(result):

src/sage/rings/polynomial/multi_polynomial_libsingular.pyx

@@ -1877,6 +1877,7 @@ cdef class MPolynomialRing_libsingular(MPolynomialRing_base):
             M.append(new_MP(self, p_Copy(tempvector, _ring)))
         return M
 
+
 def unpickle_MPolynomialRing_libsingular(base_ring, names, term_order):
     """
     inverse function for ``MPolynomialRing_libsingular.__reduce__``

src/sage/rings/polynomial/polynomial_compiled.pyx

@@ -232,8 +232,6 @@ cdef class CompiledPolynomialFunction:
 
         The r == 0 case in step 3 is equivalent to binary exponentiation.
         """
-
-
         cdef int m,n,k,r,half
         cdef generic_pd T,N,H
         cdef dummy_pd M

src/sage/rings/polynomial/polynomial_element.pyx

@@ -11554,7 +11554,6 @@ cdef class Polynomial(CommutativePolynomial):
             phi = SpecializationMorphism(self._parent,D)
         return phi(self)
 
-
     def _log_series(self, long n):
         r"""
         Return the power series expansion of logarithm of this polynomial,
@@ -11711,6 +11710,7 @@ cdef class Polynomial(CommutativePolynomial):
         """
         raise NotImplementedError
 
+
 # ----------------- inner functions -------------
 # Cython can't handle function definitions inside other function
 
@@ -11766,6 +11766,7 @@ cdef list do_schoolbook_product(list x, list y, Py_ssize_t deg):
         coeffs[k] = sum
     return coeffs
 
+
 @cython.boundscheck(False)
 @cython.wraparound(False)
 @cython.overflowcheck(False)
@@ -11848,6 +11849,7 @@ cdef list do_karatsuba_different_size(list left, list right, Py_ssize_t K_thresh
             output.extend(carry[n-1:])
         return output
 
+
 @cython.boundscheck(False)
 @cython.wraparound(False)
 @cython.overflowcheck(False)
@@ -12581,6 +12583,7 @@ cdef class Polynomial_generic_dense(Polynomial):
         self._coeffs = self._coeffs[:n]
         return self
 
+
 def make_generic_polynomial(parent, coeffs):
     return parent(coeffs)
 

src/sage/rings/polynomial/polynomial_gf2x.pyx

@@ -310,6 +310,7 @@ def GF2X_BuildIrred_list(n):
     GF2X_BuildIrred(f, int(n))
     return [GF2(not GF2_IsZero(GF2X_coeff(f, i))) for i in range(n + 1)]
 
+
 def GF2X_BuildSparseIrred_list(n):
     """
     Return the list of coefficients of an irreducible polynomial of
@@ -330,6 +331,7 @@ def GF2X_BuildSparseIrred_list(n):
     GF2X_BuildSparseIrred(f, int(n))
     return [GF2(not GF2_IsZero(GF2X_coeff(f, i))) for i in range(n + 1)]
 
+
 def GF2X_BuildRandomIrred_list(n):
     """
     Return the list of coefficients of an irreducible polynomial of

src/sage/rings/polynomial/polynomial_number_field.pyx

@@ -63,20 +63,21 @@ We can also construct polynomials over relative number fields::
     1
 """
 
-#*****************************************************************************
+# ****************************************************************************
 #       Copyright (C) 2014 Luis Felipe Tabera Alonso <[email protected]>
 #
 #  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.rings.polynomial.polynomial_element_generic import Polynomial_generic_dense_field
 from sage.rings.rational_field import QQ
 from sage.structure.element import coerce_binop
 from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
 
+
 class Polynomial_absolute_number_field_dense(Polynomial_generic_dense_field):
     """
     Class of dense univariate polynomials over an absolute number field.

src/sage/rings/polynomial/polynomial_rational_flint.pyx

@@ -165,7 +165,6 @@ cdef class Polynomial_rational_flint(Polynomial):
             fmpq_poly_set_si(res._poly, int(x))
         return res
 
-
     def __cinit__(self):
         """
         Initialises the underlying data structure.

src/sage/rings/polynomial/weil/weil_polynomials.pyx

@@ -231,6 +231,7 @@ cdef class dfs_manager:
             raise RuntimeError("Node limit ({0:%d}) exceeded".format(self.node_limit))
         return ans
 
+
 class WeilPolynomials_iter():
     r"""
     Iterator created by WeilPolynomials.

src/sage/rings/power_series_mpoly.pyx

@@ -116,7 +116,6 @@ cdef class PowerSeries_mpoly(PowerSeries):
                                  prec = prec,
                                  check =True)
 
-
     def __iter__(self):
         """
         Return an iterator over the coefficients of this power series.

src/sage/rings/power_series_ring_element.pyx

@@ -794,7 +794,6 @@ cdef class PowerSeries(AlgebraElement):
             s += " + %s"%bigoh
         return s.lstrip(" ")
 
-
     def truncate(self, prec=infinity):
         """
         The polynomial obtained from power series by truncation.
@@ -2725,7 +2724,6 @@ cdef class PowerSeries(AlgebraElement):
         """
         return multi_derivative(self, args)
 
-
     def __setitem__(self, n, value):
         """
         Called when an attempt is made to change a power series.

src/sage/rings/rational.pyx

@@ -82,6 +82,7 @@ from sage.structure.richcmp cimport rich_to_bool_sgn
 
 RealNumber_classes = ()
 
+
 def _register_real_number_class(cls):
     r"""
     Register ``cls``.
@@ -1580,7 +1581,6 @@ cdef class Rational(sage.structure.element.FieldElement):
         a, b = K.pari_bnf(proof=proof).bnfisnorm(self, flag=extra_primes)
         return K(a), Rational(b)
 
-
     def is_perfect_power(self, expected_value=False):
         r"""
         Return ``True`` if ``self`` is a perfect power.
@@ -3560,7 +3560,6 @@ cdef class Rational(sage.structure.element.FieldElement):
         else:
             return sage.rings.infinity.infinity
 
-
     def multiplicative_order(self):
         """
         Return the multiplicative order of ``self``.
@@ -3633,10 +3632,9 @@ cdef class Rational(sage.structure.element.FieldElement):
         """
         return True
 
-    #Function alias for checking if the number is a integer.  Added to solve issue 15500
+    # Function alias for checking if the number is a integer.  Added to solve issue 15500
     is_integer = is_integral
 
-
     def is_S_integral(self, S=[]):
         r"""
         Determine if the rational number is ``S``-integral.

src/sage/rings/sum_of_squares.pyx

@@ -130,6 +130,7 @@ cdef int three_squares_c(uint_fast32_t n, uint_fast32_t res[3]) noexcept:
 
     return 1
 
+
 def two_squares_pyx(uint32_t n):
     r"""
     Return a pair of non-negative integers ``(i,j)`` such that `i^2 + j^2 = n`.
@@ -184,7 +185,8 @@ def two_squares_pyx(uint32_t n):
         return (integer.smallInteger(i[0]), integer.smallInteger(i[1]))
     sig_off()
 
-    raise ValueError("%d is not a sum of 2 squares"%n)
+    raise ValueError("%d is not a sum of 2 squares" % n)
+
 
 def is_sum_of_two_squares_pyx(uint32_t n):
     r"""
@@ -214,6 +216,7 @@ def is_sum_of_two_squares_pyx(uint32_t n):
         sig_off()
         return False
 
+
 def three_squares_pyx(uint32_t n):
     r"""
     If ``n`` is a sum of three squares return a 3-tuple ``(i,j,k)`` of Sage integers
@@ -266,7 +269,8 @@ def three_squares_pyx(uint32_t n):
         return (integer.smallInteger(i[0]), integer.smallInteger(i[1]), integer.smallInteger(i[2]))
     sig_off()
 
-    raise ValueError("%d is not a sum of 3 squares"%n)
+    raise ValueError("%d is not a sum of 3 squares" % n)
+
 
 def four_squares_pyx(uint32_t n):
     r"""