Introduce algebraic closure

src/sage/symbolic/expression_conversion_algebraic.py

@@ -44,6 +44,10 @@ def __init__(self, field):
             tan
         """
         self.field = field
+        try:
+            self.field_ac = field.algebraic_closure()
+        except:
+            self.field_ac = field
 
         from sage.functions.all import reciprocal_trig_functions
         self.reciprocal_trig_functions = reciprocal_trig_functions
@@ -105,15 +109,15 @@ def arithmetic(self, ex, operator):
             if operator is pow:
                 from sage.rings.rational import Rational
                 base, expt = ex.operands()
-                base = self.field(base)
+                base = self.field_ac(base)
                 expt = Rational(expt)
                 return self.field(base**expt)
-            else:
-                if operator is add_vararg:
-                    operator = add
-                elif operator is mul_vararg:
-                    operator = mul
-                return reduce(operator, map(self, ex.operands()))
+            if operator is add_vararg:
+                operator = add
+            elif operator is mul_vararg:
+                operator = mul
+            converter = AlgebraicConverter(self.field_ac)
+            return self.field(reduce(operator, map(converter, ex.operands())))
         except TypeError:
             pass
 
@@ -251,7 +255,7 @@ def composition(self, ex, operator):
             interval = cr._get_interval()._sage_()
             return self.field.polynomial_root(poly, interval)
         elif operand is not None:
-            res = func(operand._algebraic_(self.field))
+            res = func(operand._algebraic_(self.field_ac))
             # We have to handle the case where we get the same symbolic
             # expression back.  For example, QQbar(zeta(7)).  See
             # issue #12665.