conversion of number field elts with real/complex embeddings to algebraic numbers

Author: mezzarobba

src/sage/rings/number_field/number_field_element.pyx

@@ -2316,6 +2316,43 @@ cdef class NumberFieldElement(FieldElement):
         ZZX_getitem_as_mpz(num.value, &self.__numerator, 0)
         return num / (<IntegerRing_class>ZZ)._coerce_ZZ(&self.__denominator)
 
+    def _algebraic_(self, parent):
+        r"""
+        Convert this element to an algebraic number, if possible.
+
+        EXAMPLES::
+
+            sage: NF.<alpha> = NumberField(x^5 + 7*x + 3, embedding=CC(0,1))
+            sage: QQbar(alpha)
+            -1.032202770009288? + 1.168103873894207?*I
+            sage: AA(alpha)
+            Traceback (most recent call last):
+            ...
+            ValueError: Cannot coerce algebraic number with non-zero imaginary
+            part to algebraic real
+
+            sage: NF.<alpha> = NumberField(x^5 + 7*x + 3)
+            sage: QQbar(alpha)
+            Traceback (most recent call last):
+            ...
+            ValueError: need a real or complex embedding to convert number
+            field element to algebraic number
+
+        TESTS::
+
+            sage: C.<z> = CyclotomicField(7)
+            sage: a = 2*z^2 + 5*z^4
+            sage: E = C.algebraic_closure()
+            sage: E(a)
+            -4.949886207424724? - 0.2195628712241434?*I
+        """
+        emb = self._parent.coerce_embedding()
+        if emb is None:
+            raise ValueError("need a real or complex embedding to convert "
+                             "number field element to algebraic number")
+        emb = number_field.refine_embedding(emb, infinity)
+        return parent(emb(self))
+
     def _symbolic_(self, SR):
         """
         If an embedding into CC is specified, then a representation of this

src/sage/rings/number_field/number_field_element_quadratic.pyx

@@ -1371,6 +1371,29 @@ cdef class NumberFieldElement_quadratic(NumberFieldElement_absolute):
             mpq_canonicalize(res.value)
             return res
 
+    def _algebraic_(self, parent):
+        r"""
+        Convert this element to an algebraic number, if possible.
+
+        EXAMPLES::
+
+            sage: NF.<i> = QuadraticField(-1)
+            sage: QQbar(1+i)
+            I + 1
+            sage: NF.<sqrt3> = QuadraticField(2)
+            sage: AA(sqrt3)
+            1.414213562373095?
+        """
+        import sage.rings.qqbar as qqbar
+        if (parent is qqbar.QQbar
+                and list(self._parent.polynomial()) == [1, 0, 1]):
+            # AlgebraicNumber.__init__ does a better job than
+            # NumberFieldElement._algebraic_ in this case, but
+            # QQbar._element_constructor_ calls the latter first.
+            return qqbar.AlgebraicNumber(self)
+        else:
+            return NumberFieldElement._algebraic_(self, parent)
+
     cpdef bint is_one(self):
         r"""
         Check whether this number field element is `1`.

src/sage/rings/qqbar.py

@@ -121,7 +121,7 @@
     sage: AA(I)
     Traceback (most recent call last):
     ...
-    TypeError: Illegal initializer for algebraic number
+    ValueError: Cannot coerce algebraic number with non-zero imaginary part to algebraic real
     sage: QQbar(I * golden_ratio)
     1.618033988749895?*I
     sage: AA(golden_ratio)^2 - AA(golden_ratio)