fix bad Frac(sparse polynomial ring over finite field)

The construction of the fraction field of a polynomial ring R over a
small finite fields tried ot use a class dedicated to the dense case
regardless whether R was dense or sparse, resulting in corrupted
elements represented as dense polynomials but with a parent set to a
sparse ring.

In passing, add support for fraction fields of polynomial rings over
prime fields based on NTL.

Fixes sagemath#37374

src/sage/rings/fraction_field_FpT.pyx

@@ -44,16 +44,26 @@ class FpT(FractionField_1poly_field):
         """
         INPUT:
 
-        - ``R`` -- A polynomial ring over a finite field of prime order `p` with `2 < p < 2^16`
+        - ``R`` -- A dense polynomial ring over a finite field of prime order `p` with `2 < p < 2^16`
 
         EXAMPLES::
 
             sage: R.<x> = GF(31)[]
             sage: K = R.fraction_field(); K
             Fraction Field of Univariate Polynomial Ring in x over Finite Field of size 31
+
+        TESTS::
+
+            sage: from sage.rings.fraction_field_FpT import FpT
+            sage: FpT(PolynomialRing(GF(37), ['x'], sparse=True))
+            Traceback (most recent call last):
+            ...
+            TypeError: unsupported polynomial ring
         """
         cdef long p = R.base_ring().characteristic()
         assert 2 < p < FpT.INTEGER_LIMIT
+        if not issubclass(R.element_class, Polynomial_zmod_flint):
+            raise TypeError("unsupported polynomial ring")
         self.p = p
         self.poly_ring = R
         FractionField_1poly_field.__init__(self, R, element_class=FpTElement)

src/sage/rings/polynomial/polynomial_modn_dense_ntl.pyx

@@ -852,14 +852,6 @@ cdef class Polynomial_dense_modn_ntl_zz(Polynomial_dense_mod_n):
             sage: (x-1)^5
             x^5 + 95*x^4 + 10*x^3 + 90*x^2 + 5*x + 99
 
-        Negative powers will not work::
-
-            sage: R.<x> = PolynomialRing(Integers(101), implementation='NTL')
-            sage: (x-1)^(-5)
-            Traceback (most recent call last):
-            ...
-            NotImplementedError: Fraction fields not implemented for this type.
-
         We define ``0^0`` to be unity, :issue:`13895`::
 
             sage: R.<x> = PolynomialRing(Integers(100), implementation='NTL')
@@ -872,6 +864,21 @@ cdef class Polynomial_dense_modn_ntl_zz(Polynomial_dense_mod_n):
             sage: type(R(0)^0) == type(R(0))
             True
 
+        Negative powers work (over prime fields) but use a generic
+        implementation of fraction fields::
+
+            sage: R.<x> = PolynomialRing(Integers(101), implementation='NTL')
+            sage: f = (x-1)^(-5)
+            sage: type(f)
+            <class 'sage.rings.fraction_field_element.FractionFieldElement_1poly_field'>
+            sage: (f + 2).numerator()
+            2*x^5 + 91*x^4 + 20*x^3 + 81*x^2 + 10*x + 100
+
+            sage: R.<x> = PolynomialRing(Integers(100), implementation='NTL')
+            sage: (x-1)^(-5)
+            Traceback (most recent call last):
+            ...
+            TypeError: self must be an integral domain.
         """
         cdef bint recip = 0, do_sig
         cdef long e = ee

src/sage/rings/polynomial/polynomial_ring.py

@@ -2460,10 +2460,9 @@ def fraction_field(self):
 
         EXAMPLES::
 
-            sage: R.<t> = GF(5)[]
-            sage: R.fraction_field()
-            Fraction Field of Univariate Polynomial Ring in t
-             over Finite Field of size 5
+            sage: QQbar['x'].fraction_field()
+            Fraction Field of Univariate Polynomial Ring in x over Algebraic
+            Field
 
         TESTS:
 
@@ -2483,17 +2482,14 @@ def fraction_field(self):
             sage: t(x)
             x
 
+        Issue :issue:`37374`:
+
+            sage: x = PolynomialRing(GF(37), ['x'], sparse=True).fraction_field().gen()
+            sage: type(x.numerator())
+            <class 'sage.rings.polynomial.polynomial_ring.PolynomialRing_field_with_category.element_class'>
+            sage: (x^8 + 16*x^6 + 4*x^4 + x^2 + 12).numerator() - 1
+            x^8 + 16*x^6 + 4*x^4 + x^2 + 11
         """
-        R = self.base_ring()
-        p = R.characteristic()
-        if p != 0 and R.is_prime_field():
-            try:
-                from sage.rings.fraction_field_FpT import FpT
-            except ImportError:
-                pass
-            else:
-                if 2 < p and p < FpT.INTEGER_LIMIT:
-                    return FpT(self)
         from sage.rings.fraction_field import FractionField_1poly_field
         return FractionField_1poly_field(self)
 
@@ -3566,6 +3562,31 @@ def irreducible_element(self, n, algorithm=None):
         # No suitable algorithm found, try algorithms from the base class.
         return PolynomialRing_dense_finite_field.irreducible_element(self, n, algorithm)
 
+    @cached_method
+    def fraction_field(self):
+        """
+        Returns the fraction field of self.
+
+        EXAMPLES::
+
+            sage: R.<t> = GF(5)[]
+            sage: R.fraction_field()
+            Fraction Field of Univariate Polynomial Ring in t
+             over Finite Field of size 5
+        """
+        try:
+            from sage.rings.fraction_field_FpT import FpT
+            from sage.rings.polynomial.polynomial_zmod_flint import Polynomial_zmod_flint
+        except ImportError:
+            pass
+        else:
+            p = self.base_ring().characteristic()
+            if (issubclass(self.element_class, Polynomial_zmod_flint)
+                    and 2 < p < FpT.INTEGER_LIMIT):
+                return FpT(self)
+        return super().fraction_field()
+
+
 def polygen(ring_or_element, name="x"):
     """
     Return a polynomial indeterminate.