gh-37367: improve random sampling of quotient-ring elements
    
Currently, random sampling in generic quotient rings is restricted to a
very small (and special) subset of the elements:
```sage
sage: R.<x,y> = QQ[]
sage: S = R.quotient([x^3, y^2])
sage: {S.random_element() for _ in range(999)}
{-2, -1, 0, 1, 2}
```

In this patch we add an implementation of `.random_element()` which
simply calls the `.random_element()` method of the cover ring and maps
the result to the quotient. This is still far from perfect for many
kinds of quotient rings, but it's definitely an improvement compared to
the current behavior.
    
URL: #37367
Reported by: Lorenz Panny
Reviewer(s): Giacomo Pope

Author: Release Manager

src/sage/rings/quotient_ring.py

@@ -1354,6 +1354,30 @@ def term_order(self):
         """
         return self.__R.term_order()
 
+    def random_element(self):
+        r"""
+        Return a random element of this quotient ring obtained by
+        sampling a random element of the cover ring and reducing
+        it modulo the defining ideal.
+
+        EXAMPLES::
+
+            sage: R.<x,y> = QQ[]
+            sage: S = R.quotient([x^3, y^2])
+            sage: S.random_element()  # random
+            -8/5*xbar^2 + 3/2*xbar*ybar + 2*xbar - 4/23
+
+        TESTS:
+
+        Make sure we are not just getting images of integers in this
+        ring (which would be the case if the default implementation
+        of this method was inherited from generic rings)::
+
+            sage: any(S.random_element() not in ZZ for _ in range(999))
+            True
+        """
+        return self.retract(self.cover_ring().random_element())
+
 
 class QuotientRing_generic(QuotientRing_nc, ring.CommutativeRing):
     r"""