Uniformize headlines

Author: kwankyu

src/doc/en/reference/noncommutative_polynomial_rings/index.rst

@@ -1,8 +1,8 @@
 Noncommutative Polynomials
 ==========================
 
-Univariate Ore polynomial rings
--------------------------------
+Univariate Ore Polynomials
+--------------------------
 
 .. toctree::
    :maxdepth: 1
@@ -13,13 +13,6 @@ Univariate Ore polynomial rings
    sage/rings/polynomial/skew_polynomial_element
    sage/rings/polynomial/skew_polynomial_finite_order
    sage/rings/polynomial/skew_polynomial_finite_field
-
-Fraction field of Ore polynomial rings
---------------------------------------
-
-.. toctree::
-   :maxdepth: 1
-
    sage/rings/polynomial/ore_function_field
    sage/rings/polynomial/ore_function_element
 

src/sage/rings/polynomial/ore_function_element.py

@@ -1,12 +1,11 @@
 r"""
-An element in the fraction field of a Ore polynomial ring.
+Fraction field elements of Ore polynomial rings
 
 AUTHOR:
 
 - Xavier Caruso (2020-05)
 """
 
-
 # ***************************************************************************
 #    Copyright (C) 2020 Xavier Caruso <[email protected]>
 #
@@ -17,7 +16,6 @@
 #                  https://www.gnu.org/licenses/
 # ***************************************************************************
 
-
 from sage.structure.richcmp import richcmp, op_EQ, op_NE
 from sage.misc.cachefunc import cached_method
 from sage.misc.latex import latex

src/sage/rings/polynomial/ore_function_field.py

@@ -1,5 +1,5 @@
 r"""
-Fraction fields of Ore polynomial rings.
+Fraction fields of Ore polynomial rings
 
 Sage provides support for building the fraction field of any Ore
 polynomial ring and performing basic operations in it.

src/sage/rings/polynomial/ore_polynomial_element.pyx

@@ -1,5 +1,5 @@
 r"""
-Univariate Ore Polynomials
+Univariate Ore polynomials
 
 This module provides the
 :class:`~sage.rings.polynomial.skew_polynomial_element.OrePolynomial`,

src/sage/rings/polynomial/ore_polynomial_ring.py

@@ -1,5 +1,5 @@
 r"""
-Univariate Ore Polynomial Rings
+Univariate Ore polynomial rings
 
 This module provides the
 :class:`~sage.rings.polynomial.ore_polynomial_ring.OrePolynomialRing`,

src/sage/rings/polynomial/plural.pyx

@@ -1,5 +1,5 @@
 r"""
-Noncommutative Polynomials via libSINGULAR/Plural
+Noncommutative polynomials via libSINGULAR/Plural
 
 This module provides specialized and optimized implementations for
 noncommutative multivariate polynomials over many coefficient rings, via the

src/sage/rings/polynomial/skew_polynomial_element.pyx

@@ -1,5 +1,5 @@
 r"""
-Univariate Skew Polynomials
+Univariate skew polynomials
 
 This module provides the
 :class:`~sage.rings.polynomial.skew_polynomial_element.SkewPolynomial`.

src/sage/rings/polynomial/skew_polynomial_finite_field.pyx

@@ -1,7 +1,7 @@
 r"""
-Univariate Dense Skew Polynomials over Finite Fields
+Univariate dense skew polynomials over finite fields
 
-This module provides the 
+This module provides the
 class:`~sage.rings.polynomial.skew_polynomial_finite_field.SkewPolynomial_finite_field_dense`,
 which constructs a single univariate skew polynomial over a finite field
 equipped with the Frobenius endomorphism. Among other things, it implements
@@ -336,7 +336,7 @@ cdef class SkewPolynomial_finite_field_dense(SkewPolynomial_finite_order_dense):
             ....:     N = R._reduced_norm_factor_uniform()
             ....:     counts[N] += 1
             sage: counts   # random
-            {z + 1: 969, z + 2: 31} 
+            {z + 1: 969, z + 2: 31}
         """
         skew_ring = self._parent
         F = self._reduced_norm_factored()
@@ -390,7 +390,7 @@ cdef class SkewPolynomial_finite_field_dense(SkewPolynomial_finite_order_dense):
             sage: for D in rightdiv:
             ....:     assert P.is_right_divisible_by(D), "not right divisible"
             ....:     assert D.is_irreducible(), "not irreducible"
-        
+
             sage: P = S.random_element(degree=10)
             sage: leftdiv = [ f for f in P.left_irreducible_divisors() ]   # indirect doctest
             sage: len(leftdiv) == P.count_irreducible_divisors()
@@ -401,7 +401,7 @@ cdef class SkewPolynomial_finite_field_dense(SkewPolynomial_finite_order_dense):
             ....:     assert P.is_left_divisible_by(D), "not left divisible"
             ....:     assert D.is_irreducible(), "not irreducible"
         """
-        cdef SkewPolynomial_finite_field_dense NS, P, Q, R, P1, Q1, L, V, g, d 
+        cdef SkewPolynomial_finite_field_dense NS, P, Q, R, P1, Q1, L, V, g, d
         cdef Py_ssize_t i, m, degrandom
         if not self:
             return
@@ -697,8 +697,8 @@ cdef class SkewPolynomial_finite_field_dense(SkewPolynomial_finite_order_dense):
 
         .. NOTE::
 
-            One can prove that there are always as many left 
-            irreducible monic divisors as right irreducible 
+            One can prove that there are always as many left
+            irreducible monic divisors as right irreducible
             monic divisors.
 
         EXAMPLES::
@@ -936,7 +936,7 @@ cdef class SkewPolynomial_finite_field_dense(SkewPolynomial_finite_order_dense):
 
         INPUT:
 
-        - ``uniform`` -- a boolean (default: ``False``); whether the 
+        - ``uniform`` -- a boolean (default: ``False``); whether the
           output irreducible divisor should be uniformly distributed
           among all possibilities
 
@@ -966,7 +966,7 @@ cdef class SkewPolynomial_finite_field_dense(SkewPolynomial_finite_order_dense):
             sage: F.value() == a
             True
 
-        There is a priori no guarantee on the distribution of the 
+        There is a priori no guarantee on the distribution of the
         factorizations we get. Passing in the keyword ``uniform=True``
         ensures the output is uniformly distributed among all
         factorizations::
@@ -1077,9 +1077,9 @@ cdef class SkewPolynomial_finite_field_dense(SkewPolynomial_finite_order_dense):
             ....:     assert F.value() == a, "factorization has a different value"
             ....:     for d,_ in F:
             ....:         assert d.is_irreducible(), "a factor is not irreducible"
-        
-        Note that the algorithm used in this method is probabilistic. 
-        As a consequence, if we call it two times with the same input, 
+
+        Note that the algorithm used in this method is probabilistic.
+        As a consequence, if we call it two times with the same input,
         we can get different orderings::
 
             sage: factorizations2 = [ F for F in a.factorizations() ]

src/sage/rings/polynomial/skew_polynomial_finite_order.pyx

@@ -1,5 +1,5 @@
 r"""
-Univariate Dense Skew Polynomials over a field equipped with a finite order automorphism
+Univariate dense skew polynomials over a field with a finite order automorphism
 
 AUTHOR::
 
@@ -190,13 +190,13 @@ cdef class SkewPolynomial_finite_order_dense(SkewPolynomial_generic_dense):
 
         By default, the name of the central variable is usually ``z`` (see
         :meth:`~sage.rings.polynomial.skew_polynomial_ring.SkewPolynomialRing_finite_order.center`
-        for more details about this). 
-        However, the user can specify a different variable name if desired:: 
+        for more details about this).
+        However, the user can specify a different variable name if desired::
 
             sage: a.reduced_trace(var='u')
             3*u + 4
 
-        When passing in ``var=False``, a tuple of coefficients (instead of 
+        When passing in ``var=False``, a tuple of coefficients (instead of
         an actual polynomial) is returned::
 
             sage: a.reduced_trace(var=False)
@@ -265,20 +265,20 @@ cdef class SkewPolynomial_finite_order_dense(SkewPolynomial_generic_dense):
 
         By default, the name of the central variable is usually ``z`` (see
         :meth:`~sage.rings.polynomial.skew_polynomial_ring.SkewPolynomialRing_finite_order.center`
-        for more details about this). 
+        for more details about this).
         However, the user can specify a different variable name if desired::
 
             sage: a.reduced_norm(var='u')
             u^3 + 4*u^2 + 4
 
-        When passing in ``var=False``, a tuple of coefficients (instead of 
+        When passing in ``var=False``, a tuple of coefficients (instead of
         an actual polynomial) is returned::
 
             sage: a.reduced_norm(var=False)
             (4, 0, 4, 1)
 
         TESTS:
-    
+
         We check that `N` is a multiple of `a`::
 
             sage: S(N).is_right_divisible_by(a)
@@ -320,7 +320,7 @@ cdef class SkewPolynomial_finite_order_dense(SkewPolynomial_generic_dense):
         """
         if self._norm is None:
             if self.is_zero():
-                self._norm = 0 
+                self._norm = 0
             else:
                 parent = self._parent
                 section = parent._embed_constants.section()
@@ -387,7 +387,7 @@ cdef class SkewPolynomial_finite_order_dense(SkewPolynomial_generic_dense):
         By default, the name of the variable of the reduced characteristic
         polynomial is ``x`` and the name of central variable is usually ``z``
         (see :meth:`~sage.rings.polynomial.skew_polynomial_ring.SkewPolynomialRing_finite_order.center`
-        for more details about this). 
+        for more details about this).
         The user can speciify different names if desired::
 
             sage: a.reduced_charpoly(var='T')  # variable name for the caracteristic polynomial

src/sage/rings/polynomial/skew_polynomial_ring.py

@@ -1,5 +1,5 @@
 r"""
-Skew Univariate Polynomial Rings
+Univariate skew polynomial rings
 
 This module provides the
 :class:`~sage.rings.polynomial.skew_polynomial_ring.SkewPolynomialRing`.