Simplify __invert__ for polynomial quotient ring element

Author: user202729

src/sage/rings/polynomial/polynomial_quotient_ring_element.py

@@ -386,10 +386,6 @@ def __invert__(self):
         """
         Return the inverse of this element.
 
-        .. WARNING::
-
-            Only implemented when the base ring is a field.
-
         EXAMPLES::
 
             sage: R.<x> = QQ[]
@@ -414,35 +410,21 @@ def __invert__(self):
             sage: (2*y)^(-1)
             Traceback (most recent call last):
             ...
-            NotImplementedError: The base ring (=Ring of integers modulo 16) is not a field
+            NotImplementedError
 
         Check that :issue:`29469` is fixed::
 
             sage: ~S(3)
             11
         """
-        if self._polynomial.is_zero():
-            raise ZeroDivisionError("element %s of quotient polynomial ring not invertible" % self)
-        if self._polynomial.is_one():
-            return self
-
-        parent = self.parent()
-
+        P = self.parent()
         try:
-            if self._polynomial.is_unit():
-                inv_pol = self._polynomial.inverse_of_unit()
-                return parent(inv_pol)
-        except (TypeError, NotImplementedError):
-            pass
-
-        base = parent.base_ring()
-        if not base.is_field():
-            raise NotImplementedError("The base ring (=%s) is not a field" % base)
-        g, _, a = parent.modulus().xgcd(self._polynomial)
-        if g.degree() != 0:
-            raise ZeroDivisionError("element %s of quotient polynomial ring not invertible" % self)
-        c = g[0]
-        return self.__class__(self.parent(), (~c)*a, check=False)
+            return type(self)(P, self._polynomial.inverse_mod(P.modulus()), check=False)
+        except ValueError as e:
+            if e.args[0] == "Impossible inverse modulo":
+                raise ZeroDivisionError(f"element {self} of quotient polynomial ring not invertible")
+            else:
+                raise NotImplementedError
 
     def field_extension(self, names):
         r"""