sagemathgh-41189: Accept both 'implementation' and 'impl' parameters for GF()
    
<!-- ^ Please provide a concise and informative title. -->
<!-- ^ Don't put issue numbers in the title, do this in the PR
description below. -->
<!-- ^ For example, instead of "Fixes sagemath#12345" use "Introduce new method
to calculate 1 + 2". -->
<!-- v Describe your changes below in detail. -->
<!-- v Why is this change required? What problem does it solve? -->
<!-- v If this PR resolves an open issue, please link to it here. For
example, "Fixes sagemath#12345". -->

1.Changed primary parameter from impl to implementation in docstrings
2.Marked impl as deprecated (backwards compatibility)
3.Updated examples to use implementation instead of impl
4.Added examples showing both parameters work
5. clean commit history
@nbruin review this


### 📝 Checklist

<!-- Put an `x` in all the boxes that apply. -->

- [x] The title is concise and informative.
- [x] The description explains in detail what this PR is about.
- [x] I have linked a relevant issue or discussion.
- [x] I have created tests covering the changes.
- [ ] I have updated the documentation and checked the documentation
preview.

### ⌛ Dependencies

<!-- List all open PRs that this PR logically depends on. For example,
-->
<!-- - sagemath#12345: short description why this is a dependency -->
<!-- - sagemath#34567: ... -->
related to sagemath#41123
    
URL: sagemath#41189
Reported by: Adnan Nazir
Reviewer(s): Chenxin Zhong, Vincent Macri

Author: Release Manager

src/sage/categories/pushout.py

@@ -3294,7 +3294,12 @@ def __init__(self, polys, names, embeddings=None, structures=None,
                 if latex_names[i] == latex_variable_name(name):
                     latex_names[i] = None
         self.latex_names = latex_names
-        self.kwds = kwds
+        kwds_self = dict(kwds.items())
+        if 'implementation' in kwds_self:
+            assert len(self.polys) == 1
+            self.implementations = [kwds_self['implementation']]
+            del kwds_self['implementation']
+        self.kwds = kwds_self
 
     def _apply_functor(self, R):
         """
@@ -3529,11 +3534,7 @@ def merge(self, other):
         # integers to encode degrees of extensions.
         from sage.rings.integer import Integer
         kwds_self = dict(self.kwds.items())
-        if 'impl' in kwds_self:
-            del kwds_self['impl']
         kwds_other = dict(other.kwds.items())
-        if 'impl' in kwds_other:
-            del kwds_other['impl']
         if (isinstance(self.polys[0], Integer)
                 and isinstance(other.polys[0], Integer)
                 and self.embeddings == other.embeddings == [None]

src/sage/homology/homology_vector_space_with_basis.py

@@ -1444,7 +1444,7 @@ def is_GF2(R):
         sage: from sage.homology.homology_vector_space_with_basis import is_GF2
         sage: is_GF2(GF(2))
         True
-        sage: is_GF2(GF(2, impl='ntl'))
+        sage: is_GF2(GF(2, implementation='ntl'))
         True
         sage: is_GF2(GF(3))
         False

src/sage/modules/free_module_element.pyx

@@ -1085,7 +1085,7 @@ cdef class FreeModuleElement(Vector):   # abstract base class
         Create the multiplication table of `GF(4)` using GP::
 
             sage: # needs sage.libs.pari
-            sage: k.<a> = GF(4, impl='pari_ffelt')
+            sage: k.<a> = GF(4, implementation="pari_ffelt")
             sage: v = gp(vector(list(k)))
             sage: v
             [0, 1, a, a + 1]

src/sage/rings/finite_rings/element_base.pyx

@@ -738,16 +738,16 @@ cdef class FinitePolyExtElement(FiniteRingElement):
 
         EXAMPLES::
 
-            sage: k.<a> = FiniteField(9, impl='givaro', modulus='primitive')            # needs sage.libs.linbox
+            sage: k.<a> = FiniteField(9, implementation='givaro', modulus='primitive')            # needs sage.libs.linbox
             sage: a.is_square()                                                         # needs sage.libs.linbox
             False
             sage: (a**2).is_square()                                                    # needs sage.libs.linbox
             True
-            sage: k.<a> = FiniteField(4, impl='ntl', modulus='primitive')               # needs sage.libs.ntl
+            sage: k.<a> = FiniteField(4, implementation='ntl', modulus='primitive')               # needs sage.libs.ntl
             sage: (a**2).is_square()                                                    # needs sage.libs.ntl
             True
-            sage: k.<a> = FiniteField(17^5, impl='pari_ffelt', modulus='primitive')     # needs sage.libs.pari
-            sage: a.is_square()                                                         # needs sage.libs.pari
+            sage: k.<a> = FiniteField(17^5, implementation='pari_ffelt', modulus='primitive')     # needs sage.libs.pari
+            sage: a.is_square()
             False
             sage: (a**2).is_square()                                                    # needs sage.libs.pari
             True
@@ -796,7 +796,7 @@ cdef class FinitePolyExtElement(FiniteRingElement):
             3
             sage: F(4).square_root()
             2
-            sage: K = FiniteField(7^3, 'alpha', impl='pari_ffelt')
+            sage: K = FiniteField(7^3, 'alpha', implementation='pari_ffelt')
             sage: K(3).square_root()
             Traceback (most recent call last):
             ...

src/sage/rings/finite_rings/element_givaro.pyx

@@ -580,12 +580,12 @@ cdef class Cache_givaro(Cache_base):
             sage: k._cache._element_repr(a^20)
             '2*a^3 + 2*a^2 + 2'
 
-            sage: k = FiniteField(3^4,'a', impl='givaro', repr='int')
+            sage: k = FiniteField(3^4,'a', implementation='givaro', repr='int')
             sage: a = k.gen()
             sage: k._cache._element_repr(a^20)
             '74'
 
-            sage: k = FiniteField(3^4,'a', impl='givaro', repr='log')
+            sage: k = FiniteField(3^4,'a', implementation='givaro', repr='log')
             sage: a = k.gen()
             sage: k._cache._element_repr(a^20)
             '20'

src/sage/rings/finite_rings/element_ntl_gf2e.pyx

@@ -144,7 +144,7 @@ cdef class Cache_ntl_gf2e(Cache_base):
 
         TESTS::
 
-            sage: k.<a> = GF(2^8, impl='ntl')
+            sage: k.<a> = GF(2^8, implementation="ntl")
         """
         self._parent = <FiniteField?>parent
         self._zero_element = self._new()
@@ -262,10 +262,10 @@ cdef class Cache_ntl_gf2e(Cache_base):
 
         We can coerce from PARI finite field implementations::
 
-            sage: K.<a> = GF(2^19, impl='ntl')
+            sage: K.<a> = GF(2^19, implementation="ntl")
             sage: a^20
             a^6 + a^3 + a^2 + a
-            sage: M.<c> = GF(2^19, impl='pari_ffelt')
+            sage: M.<c> = GF(2^19, implementation="pari_ffelt")
             sage: K(c^20)
             a^6 + a^3 + a^2 + a
         """
@@ -501,7 +501,7 @@ cdef class FiniteField_ntl_gf2eElement(FinitePolyExtElement):
 
         EXAMPLES::
 
-            sage: k.<a> = GF(2^8, impl='ntl') # indirect doctest
+            sage: k.<a> = GF(2^8, implementation="ntl") # indirect doctest
         """
         if parent is None:
             return
@@ -998,7 +998,7 @@ cdef class FiniteField_ntl_gf2eElement(FinitePolyExtElement):
 
         EXAMPLES::
 
-            sage: k.<a> = GF(2^8, impl='ntl')
+            sage: k.<a> = GF(2^8, implementation="ntl")
             sage: b = a^3 + a
             sage: b.minpoly()
             x^4 + x^3 + x^2 + x + 1