Trac #30783: Sage thinks that I^(2/3) = -1

Revert #25218.

Rationale: #25218 makes `a^(p/q)` where a is a number field element
attempt to return a solution to `a^p = x^q` in the number field. This
makes powering inconsistent with coercions, since number fields often
(and, in the case of quadratic fields, automatically) come with a
complex embedding and complex numbers of various kinds use the principal
branch of the power function.

Among other confusing behaviors, we have
{{{
sage: I^(2/3)
I^(2/3)
sage: QQbar(I^(2/3))
0.500000000000000? + 0.866025403784439?*I
sage: QQbar(I)^(2/3)
0.500000000000000? + 0.866025403784439?*I
}}}
but
{{{
sage: I.pyobject()^(2/3)
-1
}}}
and
{{{
sage: QQi.<i> = QuadraticField(-1)
sage: i^(2/3)
-1
}}}
but
{{{
sage: QQbar(i)^(2/3)
0.500000000000000? + 0.866025403784439?*I
}}}
as well as
{{{
sage: CC((i^(1/6))^2)
0.866025403784439 + 0.500000000000000*I
sage: CC((i^(1/3)))
-1.00000000000000*I
sage: i^(1/3)*CC(i)^(1/3)
0.500000000000000 - 0.866025403784439*I
}}}

URL: https://trac.sagemath.org/30783
Reported by: mmezzarobba
Ticket author(s): Marc Mezzarobba
Reviewer(s): Sébastien Labbé

Author: Release Manager

src/sage/rings/number_field/number_field_element.pyx

@@ -2295,12 +2295,8 @@ cdef class NumberFieldElement(FieldElement):
             sage: (1+sqrt2)^-1
             sqrt2 - 1
 
-        If the exponent is not integral, attempt this operation in the NumberField:
-
-            sage: K(2)^(1/2)
-            sqrt2
-
-        If this fails, perform this operation in the symbolic ring::
+        If the exponent is not integral, perform this operation in
+        the symbolic ring::
 
             sage: sqrt2^(1/5)
             2^(1/10)
@@ -2333,38 +2329,29 @@ cdef class NumberFieldElement(FieldElement):
         if (isinstance(base, NumberFieldElement) and
             (isinstance(exp, Integer) or type(exp) is int or exp in ZZ)):
             return generic_power(base, exp)
-
-        if (isinstance(base, NumberFieldElement) and exp in QQ):
-            qqexp = QQ(exp)
-            n = qqexp.numerator()
-            d = qqexp.denominator()
+        else:
+            cbase, cexp = canonical_coercion(base, exp)
+            if not isinstance(cbase, NumberFieldElement):
+                return cbase ** cexp
+            # Return a symbolic expression.
+            # We use the hold=True keyword argument to prevent the
+            # symbolics library from trying to simplify this expression
+            # again. This would lead to infinite loops otherwise.
+            from sage.symbolic.ring import SR
             try:
-                return base.nth_root(d)**n
-            except ValueError:
+                res = QQ(base)**QQ(exp)
+            except TypeError:
                 pass
-
-        cbase, cexp = canonical_coercion(base, exp)
-        if not isinstance(cbase, NumberFieldElement):
-            return cbase ** cexp
-        # Return a symbolic expression.
-        # We use the hold=True keyword argument to prevent the
-        # symbolics library from trying to simplify this expression
-        # again. This would lead to infinite loops otherwise.
-        from sage.symbolic.ring import SR
-        try:
-            res = QQ(base)**QQ(exp)
-        except TypeError:
-            pass
-        else:
-            if res.parent() is not SR:
-                return parent(cbase)(res)
-            return res
-        sbase = SR(base)
-        if sbase.operator() is operator.pow:
-            nbase, pexp = sbase.operands()
-            return nbase.power(pexp * exp, hold=True)
-        else:
-            return sbase.power(exp, hold=True)
+            else:
+                if res.parent() is not SR:
+                    return parent(cbase)(res)
+                return res
+            sbase = SR(base)
+            if sbase.operator() is operator.pow:
+                nbase, pexp = sbase.operands()
+                return nbase.power(pexp * exp, hold=True)
+            else:
+                return sbase.power(exp, hold=True)
 
     cdef void _reduce_c_(self):
         """