Trac #14602: Symbolic expression to number fields

The ticket stands to improve the `AlgebraicConverter` in
`sage.symbolic.expression_converters` and make it works with number
fields.

As mentioned on [http://ask.sagemath.org/question/2581/convert-
expression-to-quadraticfield ask] the following fails
{{{
sage: K = QuadraticField(3)
sage: K(sqrt(3))
Traceback (most recent call last):
...
TypeError: ...
}}}
The following gives an answer with a wrong parent
{{{
sage: x = K(3)**(1/2); x
sqrt(3)
sage: a.parent()
Symbolic Ring
}}}
while it is possible to do
{{{
sage: y = K(3).sqrt(); y
a
sage: y == K.gen()
True
}}}
Finally, we hopefully have
{{{
sage: K.gen() == sqrt(3)
sqrt(3) == sqrt(3)
sage: bool(K.gen() == sqrt(3))
True
}}}

URL: https://trac.sagemath.org/14602
Reported by: vdelecroix
Ticket author(s): Dave Morris
Reviewer(s): Vincent Delecroix

Author: Release Manager

src/sage/symbolic/expression_conversions.py

@@ -1073,6 +1073,21 @@ def arithmetic(self, ex, operator):
             Traceback (most recent call last):
             ...
             TypeError: unable to convert pi^6 to Algebraic Field
+
+        Test that :trac:`14602` is fixed::
+
+            sage: K = QuadraticField(3)
+            sage: K(sqrt(3)).parent() is K
+            True
+            sage: sqrt(K(3)).parent() is K
+            True
+            sage: (K(3)^(1/2)).parent()
+            Symbolic Ring
+            sage: bool(K.gen() == K(3)^(1/2) == sqrt(K(3)) == K(sqrt(3)) == sqrt(3))
+            True
+            sage: L = QuadraticField(3, embedding=-AA(3).sqrt())
+            sage: bool(L.gen() == -sqrt(3))
+            True
         """
         # We try to avoid simplifying, because maxima's simplify command
         # can change the value of a radical expression (by changing which