gh-35358: Lighter construction of finite field elements from lists

    
### 📚 Description

When doing intensive polynomial arithmetic with the NTL implementation
the constructor with lists is called a large number of times and may
spend a lot of time constructing the vector_space and FreeModuleElement
objects.

The very common call to vector_space(map=False) is optimized to be as
cheap as possible using the already cached object.

The common case of lists of length 0 and 1 is replaced by cheaper
shortcuts.

This improves performance when doing intensive polynomial computations
over finite field extensions.

### 📝 Checklist

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]`. -->

- [x] The title is concise, informative, and self-explanatory.
- [x] The description explains in detail what this PR is about.
- [ ] I have linked a relevant issue or discussion.
- [x] I have created tests covering the changes.
- [ ] I have updated the documentation accordingly.
    
URL: #35358
Reported by: Rémy Oudompheng
Reviewer(s): Travis Scrimshaw

src/sage/rings/finite_rings/element_pari_ffelt.pyx

@@ -270,13 +270,19 @@ cdef class FiniteFieldElement_pari_ffelt(FinitePolyExtElement):
         sage: k = FiniteField(3^11, 't', impl='pari_ffelt')
         sage: k([ 0, 1/2 ])
         2*t
+        sage: k([ 0, 1/2, 0, 0, 0, 0, 0, 0, 0, -1, 0 ])
+        2*t^9 + 2*t
         sage: k([ k(0), k(1) ])
         t
         sage: k([ GF(3)(2), GF(3^5,'u')(1) ])
         t + 2
         sage: R.<x> = PolynomialRing(k)
+        sage: k([ x/x ])
+        1
         sage: k([ R(-1), x/x ])
         t + 2
+        sage: k([ R(-1), R(0), 0 ])
+        2
 
     Check that zeros are created correctly (:trac:`11685`)::
 
@@ -496,7 +502,13 @@ cdef class FiniteFieldElement_pari_ffelt(FinitePolyExtElement):
             self.construct_from(x.constant_coefficient())
 
         elif isinstance(x, list):
-            if len(x) == self._parent.degree():
+            n = len(x)
+            if n == 0:
+                self.construct_from(None)
+            elif n == 1:
+                Fp = self._parent.base_ring()
+                self.construct_from(Fp(x[0]))
+            elif n == self._parent.degree():
                 self.construct_from(self._parent.vector_space(map=False)(x))
             else:
                 Fp = self._parent.base_ring()

src/sage/rings/finite_rings/finite_field_base.pyx

@@ -1248,8 +1248,6 @@ cdef class FiniteField(Field):
             sage: all(to_V(h(c) * e) == c * to_V(e) for e in E for c in F)
             True
         """
-        from sage.modules.all import VectorSpace
-        from sage.categories.morphism import is_Morphism
         if subfield is not None:
             if base is not None:
                 raise ValueError
@@ -1259,6 +1257,13 @@ cdef class FiniteField(Field):
             deprecation(28481, "The default value for map will be changing to True.  To keep the current behavior, explicitly pass map=False.")
             map = False
 
+        if base is None and self.__vector_space is not None and not map:
+            # A very common case: return as early as possible.
+            return self.__vector_space
+
+        from sage.modules.all import VectorSpace
+        from sage.categories.morphism import is_Morphism
+
         if base is None:
             base = self.prime_subfield()
             s = self.degree()