Change behavior of algebraic reals with respect to fractional powers

src/sage/features/sagemath.py

@@ -880,7 +880,7 @@ class sage__rings__number_field(JoinFeature):
 
         sage: # needs sage.rings.number_field
         sage: AA(-1)^(1/3)
-        -1
+        0.500000000000000? + 0.866025403784439?*I
         sage: QQbar(-1)^(1/3)
         0.500000000000000? + 0.866025403784439?*I
 

src/sage/rings/number_field/number_field_element.pyx

@@ -2832,7 +2832,7 @@ cdef class NumberFieldElement(NumberFieldElement_base):
             sage: AA(alpha)
             Traceback (most recent call last):
             ...
-            ValueError: Cannot coerce algebraic number with nonzero imaginary
+            ValueError: cannot coerce algebraic number with nonzero imaginary
             part to algebraic real
 
             sage: NF.<alpha> = NumberField(x^5 + 7*x + 3)

src/sage/rings/qqbar.py

@@ -117,14 +117,16 @@
     1
     sage: QQbar((-8)^(1/3))
     1.000000000000000? + 1.732050807568878?*I
-    sage: AA((-8)^(1/3))
-    -2
     sage: QQbar((-4)^(1/4))
     1 + 1*I
+    sage: AA((-8)^(1/3))
+    Traceback (most recent call last):
+    ...
+    ValueError: cannot coerce algebraic number with nonzero imaginary part to algebraic real
     sage: AA((-4)^(1/4))
     Traceback (most recent call last):
     ...
-    ValueError: Cannot coerce algebraic number with nonzero imaginary part to algebraic real
+    ValueError: cannot coerce algebraic number with nonzero imaginary part to algebraic real
 
 The coercion, however, goes in the other direction, since not all
 symbolic expressions are algebraic numbers::
@@ -134,11 +136,10 @@
     sage: QQbar(sqrt(2) + QQbar(sqrt(3)))                                               # needs sage.symbolic
     3.146264369941973?
 
-Note the different behavior in taking roots: for ``AA`` we prefer real
-roots if they exist, but for ``QQbar`` we take the principal root::
+Note that both for ``AA`` and ``QQbar``, we take the principal root::
 
     sage: AA(-1)^(1/3)
-    -1
+    0.500000000000000? + 0.866025403784439?*I
     sage: QQbar(-1)^(1/3)
     0.500000000000000? + 0.866025403784439?*I
 
@@ -298,8 +299,8 @@
     sage: n = (rt2 + rt3)^5; n
     308.3018001722975?
     sage: sage_input(n)
-    R.<x> = AA[]
-    v1 = AA.polynomial_root(AA.common_polynomial(x^2 - 2), RIF(RR(1.4142135623730949), RR(1.4142135623730951))) + AA.polynomial_root(AA.common_polynomial(x^2 - 3), RIF(RR(1.7320508075688772), RR(1.7320508075688774)))
+    R.<x> = QQbar[]
+    v1 = QQbar.polynomial_root(AA.common_polynomial(x^2 - 2), CIF(RIF(RR(1.4142135623730949), RR(1.4142135623730951)), RIF(RR(0)))).real() + QQbar.polynomial_root(AA.common_polynomial(x^2 - 3), CIF(RIF(RR(1.7320508075688772), RR(1.7320508075688774)), RIF(RR(0)))).real()
     v2 = v1*v1
     v2*v2*v1
 
@@ -335,8 +336,8 @@
     sage: n = rt2^2
     sage: sage_input(n, verify=True)
     # Verified
-    R.<x> = AA[]
-    v = AA.polynomial_root(AA.common_polynomial(x^2 - 2), RIF(RR(1.4142135623730949), RR(1.4142135623730951)))
+    R.<x> = QQbar[]
+    v = QQbar.polynomial_root(AA.common_polynomial(x^2 - 2), CIF(RIF(RR(1.4142135623730949), RR(1.4142135623730951)), RIF(RR(0)))).real()
     v*v
     sage: sage_input(n, verify=True)
     # Verified
@@ -1157,7 +1158,7 @@
             if x.imag().is_zero():
                 return x.real()
             else:
-                raise ValueError("Cannot coerce algebraic number with nonzero imaginary part to algebraic real")
+                raise ValueError("cannot coerce algebraic number with nonzero imaginary part to algebraic real")
         elif hasattr(x, '_algebraic_'):
             return x._algebraic_(AA)
         return AlgebraicReal(x)
@@ -4324,9 +4325,8 @@
         .. WARNING::
 
             Note that for odd `n`, all ``False`` and negative real numbers,
-            ``AlgebraicReal`` and ``AlgebraicNumber`` values give different
-            answers: ``AlgebraicReal`` values prefer real results, and
-            ``AlgebraicNumber`` values return the principal root.
+            ``AlgebraicReal`` and ``AlgebraicNumber`` values return the
+            principal root.
 
         EXAMPLES::
 
@@ -4361,6 +4361,9 @@
             True
         """
         if not all:
+            if self.parent() is AA and self.sign() < 0:
+                if n % 2:
+                    return -((-self) ** ~ZZ(n))
             return self ** ~ZZ(n)
         else:
             root = QQbar(self) ** ~ZZ(n)
@@ -5101,7 +5104,7 @@
             sage: QQbar.zeta(3)._mpfr_(RR)
             Traceback (most recent call last):
             ...
-            ValueError: Cannot coerce algebraic number with nonzero imaginary part to algebraic real
+            ValueError: cannot coerce algebraic number with nonzero imaginary part to algebraic real
         """
         return AA(self)._mpfr_(field)
 
@@ -5120,7 +5123,7 @@
             sage: float(QQbar.zeta(3))
             Traceback (most recent call last):
             ...
-            ValueError: Cannot coerce algebraic number with nonzero imaginary part to algebraic real
+            ValueError: cannot coerce algebraic number with nonzero imaginary part to algebraic real
         """
         return AA(self).__float__()
 
@@ -5174,13 +5177,13 @@
             sage: QQbar.zeta(6)._integer_()
             Traceback (most recent call last):
             ...
-            ValueError: Cannot coerce algebraic number with nonzero imaginary part to algebraic real
+            ValueError: cannot coerce algebraic number with nonzero imaginary part to algebraic real
 
             sage: # needs sage.symbolic
             sage: QQbar(sqrt(17))._integer_()
             Traceback (most recent call last):
             ...
-            ValueError: Cannot coerce non-integral Algebraic Real 4.123105625617660? to Integer
+            ValueError: cannot coerce non-integral Algebraic Real 4.123105625617660? to Integer
             sage: QQbar(sqrt(16))._integer_()
             4
             sage: v = QQbar(1 + I*sqrt(3))^5 + QQbar(16*sqrt(3)*I); v
@@ -5203,13 +5206,13 @@
             sage: (QQbar.zeta(7)^3)._rational_()
             Traceback (most recent call last):
             ...
-            ValueError: Cannot coerce algebraic number with nonzero imaginary part to algebraic real
+            ValueError: cannot coerce algebraic number with nonzero imaginary part to algebraic real
 
             sage: # needs sage.symbolic
             sage: QQbar(sqrt(2))._rational_()
             Traceback (most recent call last):
             ...
-            ValueError: Cannot coerce irrational Algebraic Real 1.414213562373095? to Rational
+            ValueError: cannot coerce irrational Algebraic Real 1.414213562373095? to Rational
             sage: v1 = QQbar(1/3 + I*sqrt(5))^7
             sage: v2 = QQbar((100336/729*golden_ratio - 50168/729)*I)
             sage: v = v1 + v2; v
@@ -5654,21 +5657,21 @@
             sage: AA(golden_ratio)._integer_()                                          # needs sage.symbolic
             Traceback (most recent call last):
             ...
-            ValueError: Cannot coerce non-integral Algebraic Real 1.618033988749895? to Integer
+            ValueError: cannot coerce non-integral Algebraic Real 1.618033988749895? to Integer
             sage: (AA(golden_ratio)^10 + AA(1-golden_ratio)^10)._integer_()             # needs sage.symbolic
             123
             sage: AA(-22/7)._integer_()
             Traceback (most recent call last):
             ...
-            ValueError: Cannot coerce non-integral Algebraic Real -22/7 to Integer
+            ValueError: cannot coerce non-integral Algebraic Real -22/7 to Integer
         """
         if self._value.lower().ceiling() > self._value.upper().floor():
             # The value is known to be non-integral.
-            raise ValueError(lazy_string("Cannot coerce non-integral Algebraic Real %s to Integer", self))
+            raise ValueError(lazy_string("cannot coerce non-integral Algebraic Real %s to Integer", self))
 
         self.exactify()
         if not isinstance(self._descr, ANRational):
-            raise ValueError(lazy_string("Cannot coerce irrational Algebraic Real %s to Integer", self))
+            raise ValueError(lazy_string("cannot coerce irrational Algebraic Real %s to Integer", self))
 
         return ZZ(self._descr._value)
 
@@ -5790,15 +5793,15 @@
             sage: AA(sqrt(7))._rational_()                                              # needs sage.symbolic
             Traceback (most recent call last):
             ...
-            ValueError: Cannot coerce irrational Algebraic Real 2.645751311064591? to Rational
+            ValueError: cannot coerce irrational Algebraic Real 2.645751311064591? to Rational
             sage: v = AA(1/2 + sqrt(2))^3 - AA(11/4*sqrt(2)); v                         # needs sage.symbolic
             3.125000000000000?
             sage: v._rational_()                                                        # needs sage.symbolic
             25/8
         """
         self.exactify()
         if not isinstance(self._descr, ANRational):
-            raise ValueError(lazy_string("Cannot coerce irrational Algebraic Real %s to Rational", self))
+            raise ValueError(lazy_string("cannot coerce irrational Algebraic Real %s to Rational", self))
 
         return QQ(self._descr._value)
 
@@ -6373,9 +6376,9 @@
     TESTS::
 
         sage: AA(-8)^(1/3)
-        -2
+        1.000000000000000? + 1.732050807568878?*I
         sage: AA(-8)^(2/3)
-        4
+        -2.000000000000000? + 3.464101615137755?*I
         sage: AA(32)^(3/5)
         8
         sage: AA(-16)^(1/2)
@@ -6398,7 +6401,7 @@
             sage: act = AlgebraicNumberPowQQAction(QQ, AA); act
             Right Rational Powering by Rational Field on Algebraic Real Field
             sage: act(AA(-2), 1/3)
-            -1.259921049894873?
+            0.6299605249474365? + 1.091123635971722?*I
 
         ::
 
@@ -6429,19 +6432,16 @@
 
         # Parent of the result
         S = self.codomain()
-        if S is AA and d % 2 == 0 and x.sign() < 0:
+        if S is AA and x.sign() < 0:
             S = QQbar
 
-        # First, check for exact roots.
+        # First, check for exact roots
         if isinstance(x._descr, ANRational):
             rt = rational_exact_root(abs(x._descr._value), d)
             if rt is not None:
                 if x._descr._value < 0:
-                    if S is AA:
-                        return AlgebraicReal(ANRational((-rt)**n))
-                    else:
-                        z = QQbar.zeta(2 * d)._pow_int(n)
-                        return z * AlgebraicNumber(ANRational(rt**n))
+                    z = QQbar.zeta(2 * d)._pow_int(n)
+                    return z * AlgebraicNumber(ANRational(rt**n))
                 return S(ANRational(rt**n))
 
         if S is AA:

src/sage/rings/universal_cyclotomic_field.py

@@ -642,7 +642,7 @@ def _algebraic_(self, R):
             sage: AA(UCF.gen(5))
             Traceback (most recent call last):
             ...
-            ValueError: Cannot coerce algebraic number with nonzero imaginary
+            ValueError: cannot coerce algebraic number with nonzero imaginary
             part to algebraic real
         """
         return R(QQbar(self))

src/sage/structure/element.pyx

@@ -2976,6 +2976,46 @@ cdef class RingElement(ModuleElement):
             return False
         return self._parent.ideal(self).is_prime()
 
+    def pow(self, exponent):
+        """
+        Return ``self`` raised to the power of ``exponent``.
+
+        INPUT:
+
+        - ``exponent`` -- integer or rational number
+
+        If ``exponent`` is a rational number, this method returns the
+        fractional power of ``self``, computed via :meth:`nth_root`.
+
+        Unlike powering by ``**``, the returned value is always
+        an element of the parent of ``self``.
+
+        EXAMPLES:
+
+            sage: ZZ(-8).pow(2/3)
+            4
+            sage: QQ(-8).pow(2/3)
+            4
+            sage: RR(-8).pow(2/3)
+            4.00000000000000
+            sage: CC(-8).pow(2/3)
+            -2.00000000000000 + 3.46410161513776*I
+
+            sage: # needs sage.rings.number_field
+            sage: AA(-8).pow(2/3)
+            4
+            sage: QQbar(-8).pow(2/3)
+            -2.000000000000000? + 3.464101615137755?*I
+        """
+        if isinstance(exponent, int):
+            return self**exponent
+        try:
+            n = exponent.numerator()
+            d = exponent.denominator()
+        except AttributeError:
+            raise ArithmeticError("exponent must be an integer or a rational number")
+        return self.nth_root(d)**n
+
 
 def is_CommutativeRingElement(x):
     """

src/sage/symbolic/expression_conversion_algebraic.py

@@ -26,11 +26,9 @@
 from sage.symbolic.expression_conversions import Converter
 from sage.symbolic.operators import add_vararg, mul_vararg
 from sage.symbolic.ring import SR
+from sage.rings.abc import AlgebraicRealField
 
 
-#############
-# Algebraic #
-#############
 class AlgebraicConverter(Converter):
     def __init__(self, field):
         """
@@ -44,6 +42,11 @@ def __init__(self, field):
             tan
         """
         self.field = field
+        # converter for operands
+        if isinstance(field, AlgebraicRealField):
+            self.converter = AlgebraicConverter(field.algebraic_closure())
+        else:
+            self.converter = self
 
         from sage.functions.all import reciprocal_trig_functions
         self.reciprocal_trig_functions = reciprocal_trig_functions
@@ -97,6 +100,12 @@ def arithmetic(self, ex, operator):
             sage: L = QuadraticField(3, embedding=-AA(3).sqrt())
             sage: bool(L.gen() == -sqrt(3))
             True
+
+        Test that :issue:12745 is fixed:
+
+            sage: x = exp(2*I*pi/7) + exp(-2*I*pi/7)
+            sage: AA(x)
+            1.246979603717467?
         """
         # We try to avoid simplifying, because maxima's simplify command
         # can change the value of a radical expression (by changing which
@@ -105,26 +114,17 @@ 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.converter.field(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
+            return self.field(reduce(operator, map(self.converter, ex.operands())))
         except TypeError:
             pass
 
-        if operator is pow:
-            from sage.symbolic.constants import e, pi, I
-            from sage.rings.rational_field import QQ
-
-            base, expt = ex.operands()
-            if base == e and expt / (pi * I) in QQ:
-                return exp(expt)._algebraic_(self.field)
-
         raise TypeError("unable to convert %r to %s" % (ex, self.field))
 
     def composition(self, ex, operator):