Trac #23166: Conversion from a Function Field to its underlying Polyn…

…omial Ring

{{{
sage: K.<x> = FunctionField(QQ)
sage: K._ring(x)
Traceback:
...
ValueError: only constants can be converted into the constant base field
but x is not a constant
}}}

URL: https://trac.sagemath.org/23166
Reported by: saraedum
Ticket author(s): David Roe
Reviewer(s): Julian Rüth

src/sage/rings/function_field/function_field.py

@@ -1427,6 +1427,9 @@ def __init__(self, constant_field, names,
         self._ring = R
         self._field = R.fraction_field()
         self._populate_coercion_lists_(coerce_list=[self._field])
+        from sage.categories.sets_with_partial_maps import SetsWithPartialMaps
+        from sage.categories.morphism import SetMorphism
+        R.register_conversion(SetMorphism(self.Hom(R, SetsWithPartialMaps()), self._to_polynomial))
         self._gen = self(R.gen())
 
     def __reduce__(self):
@@ -1562,6 +1565,25 @@ def _to_constant_base_field(self, f):
             return K(f.numerator()) / K(f.denominator())
         raise ValueError("only constants can be converted into the constant base field but %r is not a constant"%(f,))
 
+    def _to_polynomial(self, f):
+        """
+        If ``f`` is integral, return it as a polynomial.
+
+        INPUT:
+
+        - ``f`` -- an element of this rational function field whose denominator is a constant.
+
+        EXAMPLES::
+
+            sage: K.<x> = FunctionField(QQ)
+            sage: K._ring(x) # indirect doctest
+            x
+        """
+        K = f.parent().constant_base_field()
+        if f.denominator() in K:
+            return f.numerator()/K(f.denominator())
+        raise ValueError("Only polynomials can be converted to the underlying polynomial ring")
+
     def _to_bivariate_polynomial(self, f):
         """
         Convert ``f`` from a univariate polynomial over the rational function