gh-36105: `sage.rings.finite_rings.element_*`: Remove unused imports
    
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Likewise for the module providing the parent class.

Also replace an import that is more specific than necessary, which helps
for modularization.

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- Part of: #29705
- Cherry-picked from: #35095
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URL: #36105
Reported by: Matthias Köppe
Reviewer(s): David Coudert

Author: Release Manager

src/sage/rings/finite_rings/element_givaro.pyx

@@ -65,17 +65,16 @@ from cypari2.gen cimport Gen
 from cypari2.stack cimport clear_stack
 
 from sage.structure.parent cimport Parent
+from sage.structure.element cimport Vector
 
 from sage.interfaces.abc import GapElement
 
 cdef object is_IntegerMod
 cdef object Integer
 cdef object Rational
-cdef object ConwayPolynomials
-cdef object conway_polynomial
 cdef object MPolynomial
 cdef object Polynomial
-cdef object FreeModuleElement
+
 
 cdef void late_import():
     """
@@ -84,11 +83,8 @@ cdef void late_import():
     global is_IntegerMod, \
            Integer, \
            Rational, \
-           ConwayPolynomials, \
-           conway_polynomial, \
            MPolynomial, \
-           Polynomial, \
-           FreeModuleElement
+           Polynomial
 
     if is_IntegerMod is not None:
         return
@@ -102,20 +98,12 @@ cdef void late_import():
     import sage.rings.rational
     Rational = sage.rings.rational.Rational
 
-    import sage.databases.conway
-    ConwayPolynomials = sage.databases.conway.ConwayPolynomials
-
-    import sage.rings.finite_rings.finite_field_constructor
-    conway_polynomial = sage.rings.finite_rings.conway_polynomials.conway_polynomial
-
     import sage.rings.polynomial.multi_polynomial_element
     MPolynomial = sage.rings.polynomial.multi_polynomial_element.MPolynomial
 
     import sage.rings.polynomial.polynomial_element
     Polynomial = sage.rings.polynomial.polynomial_element.Polynomial
 
-    import sage.modules.free_module_element
-    FreeModuleElement = sage.modules.free_module_element.FreeModuleElement
 
 cdef class Cache_givaro(Cache_base):
     def __init__(self, parent, unsigned int p, unsigned int k, modulus, repr="poly", cache=False):
@@ -391,7 +379,7 @@ cdef class Cache_givaro(Cache_base):
             return self.parent(eval(e.replace("^", "**"),
                                     self.parent.gens_dict()))
 
-        elif isinstance(e, FreeModuleElement):
+        elif isinstance(e, Vector):
             if self.parent.vector_space(map=False) != e.parent():
                 raise TypeError("e.parent must match self.vector_space")
             ret = self._zero_element

src/sage/rings/finite_rings/element_ntl_gf2e.pyx

@@ -36,6 +36,7 @@ from sage.structure.richcmp cimport (richcmp,
                                      richcmp_not_equal, rich_to_bool)
 
 from sage.structure.parent cimport Parent
+from sage.structure.element cimport Vector
 
 from sage.rings.finite_rings.finite_field_base cimport FiniteField
 
@@ -50,15 +51,10 @@ from sage.interfaces.abc import GapElement
 
 
 cdef object is_IntegerMod
-cdef object IntegerModRing_generic
 cdef object Integer
 cdef object Rational
-cdef object is_Polynomial
-cdef object ConwayPolynomials
-cdef object conway_polynomial
 cdef object MPolynomial
 cdef object Polynomial
-cdef object FreeModuleElement
 cdef object GF
 cdef object GF2, GF2_0, GF2_1
 
@@ -69,15 +65,10 @@ cdef int late_import() except -1:
     imports.
     """
     global is_IntegerMod, \
-           IntegerModRing_generic, \
            Integer, \
            Rational, \
-           is_Polynomial, \
-           ConwayPolynomials, \
-           conway_polynomial, \
            MPolynomial, \
            Polynomial, \
-           FreeModuleElement, \
            GF, \
            GF2, GF2_0, GF2_1
 
@@ -87,30 +78,15 @@ cdef int late_import() except -1:
     import sage.rings.finite_rings.integer_mod
     is_IntegerMod = sage.rings.finite_rings.integer_mod.is_IntegerMod
 
-    import sage.rings.finite_rings.integer_mod_ring
-    IntegerModRing_generic = sage.rings.finite_rings.integer_mod_ring.IntegerModRing_generic
-
     import sage.rings.rational
     Rational = sage.rings.rational.Rational
 
-    import sage.rings.polynomial.polynomial_element
-    is_Polynomial = sage.rings.polynomial.polynomial_element.is_Polynomial
-
-    import sage.databases.conway
-    ConwayPolynomials = sage.databases.conway.ConwayPolynomials
-
-    import sage.rings.finite_rings.conway_polynomials
-    conway_polynomial = sage.rings.finite_rings.conway_polynomials.conway_polynomial
-
     import sage.rings.polynomial.multi_polynomial_element
     MPolynomial = sage.rings.polynomial.multi_polynomial_element.MPolynomial
 
     import sage.rings.polynomial.polynomial_element
     Polynomial = sage.rings.polynomial.polynomial_element.Polynomial
 
-    import sage.modules.free_module_element
-    FreeModuleElement = sage.modules.free_module_element.FreeModuleElement
-
     import sage.rings.finite_rings.finite_field_constructor
     GF = sage.rings.finite_rings.finite_field_constructor.FiniteField
     GF2 = GF(2)
@@ -313,7 +289,7 @@ cdef class Cache_ntl_gf2e(Cache_base):
         elif isinstance(e, str):
             return self._parent(eval(e.replace("^","**"),self._parent.gens_dict()))
 
-        elif isinstance(e, FreeModuleElement):
+        elif isinstance(e, Vector):
             if self._parent.vector_space(map=False) != e.parent():
                 raise TypeError("e.parent must match self.vector_space")
             ztmp = Integer(e.list(),2)

src/sage/rings/finite_rings/element_pari_ffelt.pyx

@@ -32,15 +32,12 @@ from sage.rings.integer cimport Integer
 from sage.rings.polynomial.polynomial_element import Polynomial
 from sage.rings.polynomial.multi_polynomial_element import MPolynomial
 from sage.rings.rational import Rational
+from sage.structure.element cimport Vector
 from sage.structure.richcmp cimport rich_to_bool
 
-try:
-    from sage.modules.free_module_element import FreeModuleElement
-except ImportError:
-    FreeModuleElement = ()
-
 from sage.interfaces.abc import GapElement
 
+
 cdef GEN _INT_to_FFELT(GEN g, GEN x) except NULL:
     """
     Convert the t_INT `x` to an element of the field of definition of
@@ -457,7 +454,7 @@ cdef class FiniteFieldElement_pari_ffelt(FinitePolyExtElement):
             sig_off()
             raise TypeError(f"unable to convert PARI {x.type()} to finite field element")
 
-        elif (isinstance(x, FreeModuleElement)
+        elif (isinstance(x, Vector)
               and x.parent() is self._parent.vector_space(map=False)):
             g = (<pari_gen>self._parent._gen_pari).g
             t = g[1]  # codeword: t_FF_FpXQ, t_FF_Flxq, t_FF_F2xq

src/sage/rings/finite_rings/finite_field_ntl_gf2e.py

@@ -33,14 +33,7 @@ def late_import():
     """
     if "GF2" in globals():
         return
-    global is_FiniteField, exists_conway_polynomial, conway_polynomial, Cache_ntl_gf2e, GF, GF2, is_Polynomial
-
-    import sage.rings.finite_rings.finite_field_base
-    is_FiniteField = sage.rings.finite_rings.finite_field_base.is_FiniteField
-
-    import sage.rings.finite_rings.conway_polynomials
-    exists_conway_polynomial = sage.rings.finite_rings.conway_polynomials.exists_conway_polynomial
-    conway_polynomial = sage.rings.finite_rings.conway_polynomials.conway_polynomial
+    global Cache_ntl_gf2e, GF, GF2
 
     import sage.rings.finite_rings.element_ntl_gf2e
     Cache_ntl_gf2e = sage.rings.finite_rings.element_ntl_gf2e.Cache_ntl_gf2e
@@ -49,9 +42,6 @@ def late_import():
     GF = sage.rings.finite_rings.finite_field_constructor.GF
     GF2 = GF(2)
 
-    import sage.rings.polynomial.polynomial_element
-    is_Polynomial = sage.rings.polynomial.polynomial_element.is_Polynomial
-
 
 class FiniteField_ntl_gf2e(FiniteField):
     """