Trac #25218: Extract roots in NumberField if possible

!NumberField previously evaluated integral powers in the !NumberField,
and evaluated all fractional powers in the symbolic ring.

This patch makes !NumberField attempt to evaluate the fractional power
within the field, and only falls back on the symbolic ring if this
fails.

There's a few interesting changes in the test suite.

Old code:

{{{
sage: QQbar((2*I)^(1/2))
1 + 1*I
sage: (2*I)^(1/2)
sqrt(2*I)
sage: I^(2/3)
I^(2/3)
}}}

New code:

{{{
sage: QQbar((2*I)^(1/2))
I + 1
sage: (2*I)^(1/2)
I + 1
sage: I^(2/3)
-1
}}}

The first change is just cosmetic.  The second makes good sense, as Sage
is now evaluating an expression it didn't before.  The third change is
more troubling.

The explanation lies in the definition of I:

{{{
sage: I.parent()
Symbolic Ring
sage: I.pyobject().parent()
Number Field in I with defining polynomial x^2 + 1
}}}

In this number field, there is a single cube root of `I` (`-I`).
Squaring `-I` gives us `-1`, so `I^(2/3)=-1`.

My opinion is that the new behavior of !NumberField is correct and
preferred, but perhaps `I` should be defined in QQbar, not in a
!NumberField.

URL: https://trac.sagemath.org/25218
Reported by: gh-BrentBaccala
Ticket author(s): Brent Baccala
Reviewer(s): Sébastien Labbé

Author: Release Manager

src/sage/rings/number_field/number_field_element.pyx

@@ -2179,8 +2179,12 @@ cdef class NumberFieldElement(FieldElement):
             sage: (1+sqrt2)^-1
             sqrt2 - 1
 
-        If the exponent is not integral, perform this operation in
-        the symbolic ring::
+        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::
 
             sage: sqrt2^(1/5)
             2^(1/10)
@@ -2213,29 +2217,38 @@ 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)
-        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
+
+        if (isinstance(base, NumberFieldElement) and exp in QQ):
+            qqexp = QQ(exp)
+            n = qqexp.numerator()
+            d = qqexp.denominator()
             try:
-                res = QQ(base)**QQ(exp)
-            except TypeError:
+                return base.nth_root(d)**n
+            except ValueError:
                 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)
+
+        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)
 
     cdef void _reduce_c_(self):
         """

src/sage/symbolic/expression.pyx

@@ -1461,7 +1461,7 @@ cdef class Expression(CommutativeRingElement):
             sage: AA(-golden_ratio)
             -1.618033988749895?
             sage: QQbar((2*I)^(1/2))
-            1 + 1*I
+            I + 1
             sage: QQbar(e^(pi*I/3))
             0.50000000000000000? + 0.866025403784439?*I
 
@@ -3902,11 +3902,11 @@ cdef class Expression(CommutativeRingElement):
             sage: I^(1/2)
             sqrt(I)
             sage: I^(2/3)
-            I^(2/3)
+            -1
             sage: 2^(1/2)
             sqrt(2)
             sage: (2*I)^(1/2)
-            sqrt(2*I)
+            I + 1
 
         Test if we can take powers of elements of `\QQ(i)` (:trac:`8659`)::
 

src/sage/symbolic/expression_conversions.py

@@ -1058,7 +1058,7 @@ def algebraic(ex, field):
         sage: AA(-golden_ratio)
         -1.618033988749895?
         sage: QQbar((2*I)^(1/2))
-        1 + 1*I
+        I + 1
         sage: QQbar(e^(pi*I/3))
         0.50000000000000000? + 0.866025403784439?*I