Changed _number_field_from_algebraics in projective point as well

Fixed numerous bugs, including handling the case where QQbar elements land in QQ

src/sage/schemes/projective/projective_morphism.py

@@ -5204,7 +5204,7 @@ def _number_field_from_algebraics(self):
             sage: P.<x,y>=ProjectiveSpace(QQbar,1)
             sage: H = Hom(P,P)
             sage: f = H([s*x^3-13*y^3, y^3-15*y^3])
-            sage: print f
+            sage: f
             Scheme endomorphism of Projective Space of dimension 1 over Algebraic Field
               Defn: Defined on coordinates by sending (x : y) to
                     ((-0.6823278038280193?)*x^3 + (-13)*y^3 : (-14)*y^3)
@@ -5220,10 +5220,16 @@ def _number_field_from_algebraics(self):
             raise NotImplementedError("not implemented for subschemes")
 
         K_pre,C,phi = number_field_elements_from_algebraics([c for f in self for c in f.coefficients()])
-        # The field K_pre returned above does not have its embedding set, so we redefine it:
-        K = NumberField(K_pre.polynomial(), embedding=phi(K_pre.gen()), name='b')
-        psi = K_pre.hom([K.gen()], K)
-        C = [ psi(c) for c in C ]
+        # Trac 23808: The field K_pre returned above does not have its embedding set to be phi
+        # and phi is forgotten, so we redefine K_pre to be a field K with phi as the specified
+        # embedding:
+        if K_pre is QQ:
+            K = QQ
+        else:
+            from sage.rings.number_field.number_field import NumberField
+            K = NumberField(K_pre.polynomial(), embedding=phi(K_pre.gen()), name='a')
+            psi = K_pre.hom([K.gen()], K) # Identification of K_pre with K
+            C = [ psi(c) for c in C ] # The elements of C were in K_pre, move them to K
         from sage.schemes.projective.projective_space import ProjectiveSpace
         N = self.domain().dimension_relative()
         PS = ProjectiveSpace(K,N,'z')

src/sage/schemes/projective/projective_point.py

@@ -1748,12 +1748,36 @@ def _number_field_from_algebraics(self):
             (1/2*a^3 + a^2 - 1/2*a : 1)
             sage: S.codomain()
             Projective Space of dimension 1 over Number Field in a with defining polynomial y^4 + 1
+
+        The following was fixed in :trac:`23808`::
+
+            sage: R.<x> = PolynomialRing(QQ)
+            sage: P.<x,y> = ProjectiveSpace(QQbar,1)
+            sage: Q = P([-1/2*QQbar(sqrt(2)) + QQbar(I), 1]);Q
+            (-0.7071067811865475? + 1*I : 1)
+            sage: S = Q._number_field_from_algebraics(); S
+            (1/2*a^3 + a^2 - 1/2*a : 1)
+            sage: T = S.change_ring(QQbar) # Used to fail
+            sage: T
+            (-0.7071067811865475? + 1.000000000000000?*I : 1)
+            sage: Q[0] == T[0]
+            True
         """
         from sage.schemes.projective.projective_space import is_ProjectiveSpace
         if not is_ProjectiveSpace(self.codomain()):
             raise NotImplementedError("not implemented for subschemes")
 
-        K,P,phi = number_field_elements_from_algebraics(list(self))
+        # Trac #23808: Keep the embedding info associated with the number field K
+        # used below, instead of in the separate embedding map phi which is
+        # forgotten.
+        K_pre,P,phi = number_field_elements_from_algebraics(list(self))
+        if K_pre is QQ:
+            K = QQ
+        else:
+            from sage.rings.number_field.number_field import NumberField
+            K = NumberField(K_pre.polynomial(), embedding=phi(K_pre.gen()), name='a')
+            psi = K_pre.hom([K.gen()], K) # Identification of K_pre with K
+            P = [ psi(p) for p in P ] # The elements of P were elements of K_pre
         from sage.schemes.projective.projective_space import ProjectiveSpace
         PS = ProjectiveSpace(K,self.codomain().dimension_relative(),'z')
         return(PS(P))