Raise type error for number field morphisms of non-embedded number fields

Author: simon-king-jena

src/sage/rings/number_field/number_field.py

@@ -101,6 +101,7 @@
 from sage.misc.functional import is_odd
 from sage.misc.misc_c import prod
 from sage.categories.homset import End
+from sage.categories.pushout import pushout
 
 import sage.rings.ring
 from sage.misc.latex import latex_variable_name
@@ -6142,21 +6143,26 @@ def _coerce_map_from_(self, R):
             if R.coerce_embedding() is not None:
                 if self.coerce_embedding() is not None:
                     try:
-                        from sage.categories.pushout import pushout
-                        ambient_field = pushout(R.coerce_embedding().codomain(), self.coerce_embedding().codomain())
-                        if ambient_field is not None:
-                            try:
-                                # the original ambient field
-                                return number_field_morphisms.EmbeddedNumberFieldMorphism(R, self, ambient_field)
-                            except ValueError: # no embedding found
-                                # there might be one in the alg. completion
-                                return number_field_morphisms.EmbeddedNumberFieldMorphism(R, self, ambient_field.algebraic_closure() if hasattr(ambient_field,'algebraic_closure') else ambient_field)
-                    except (ValueError, TypeError, NotImplementedError, sage.structure.coerce_exceptions.CoercionException),msg:
-                        # no success with the pushout
                         try:
                             return number_field_morphisms.EmbeddedNumberFieldMorphism(R, self)
-                        except (TypeError, ValueError):
+                        except ValueError: # no embedding found
                             pass
+                        Remb = R.coerce_embedding().codomain()
+                        while Remb.coerce_embedding() is not None:
+                            Remb = Remb.coerce_embedding().codomain()
+                        semb = self.coerce_embedding().codomain()
+                        while semb.coerce_embedding() is not None:
+                            semb = semb.coerce_embedding().codomain()
+                        ambient_field = pushout(Remb, semb)
+                        # By trac #15331, EmbeddedNumberFieldMorphism has already
+                        # tried to use this pushout. Hence, if we are here, our
+                        # only chance for success is the algebrais closure.
+                        if ambient_field is not None:
+                            return number_field_morphisms.EmbeddedNumberFieldMorphism(R, self,
+                                                                           ambient_field.algebraic_closure())
+                    except (AttributeError, ValueError, TypeError, NotImplementedError, sage.structure.coerce_exceptions.CoercionException),msg:
+                        # no success at all
+                        return
                 else:
                     # R is embedded, self isn't. So, we could only have
                     # the forgetful coercion. But this yields to non-commuting

src/sage/rings/number_field/number_field_morphisms.pyx

@@ -24,6 +24,7 @@ fields (generally `\RR` or `\CC`).
 from sage.structure.element cimport Element
 from sage.categories.morphism cimport Morphism
 from sage.categories.map cimport Map
+from sage.categories.pushout import pushout
 
 from sage.rings.real_mpfr import RealField, mpfr_prec_min
 from sage.rings.complex_field import ComplexField
@@ -107,8 +108,7 @@ cdef class NumberFieldEmbedding(Morphism):
 cdef class EmbeddedNumberFieldMorphism(NumberFieldEmbedding):
     r"""
     This allows one to go from one number field in another consistently,
-    assuming they both have specified embeddings into an ambient field
-    (by default it looks for an embedding into `\CC`).
+    assuming they both have specified embeddings into an ambient field.
 
     EXAMPLES::
 
@@ -161,13 +161,43 @@ cdef class EmbeddedNumberFieldMorphism(NumberFieldEmbedding):
             Traceback (most recent call last):
             ...
             TypeError: unsupported operand parent(s) for '+': 'Number Field in a with defining polynomial x^3 + 2' and 'Number Field in a with defining polynomial x^3 + 2'
+
+        The following was fixed to raise a ``TypeError`` in :trac:`15331`::
+
+            sage: L.<i> = NumberField(x^2 + 1)
+            sage: K = NumberField(L(i/2+3).minpoly(), names=('i0',), embedding=L(i/2+3))
+            sage: EmbeddedNumberFieldMorphism(K, L)
+            Traceback (most recent call last):
+            ...
+            TypeError: No embedding available for Number Field in i with defining polynomial x^2 + 1
+
         """
         if ambient_field is None:
             from sage.rings.complex_double import CDF
-            ambient_field = CDF
+            default_ambient_field = CDF
+            Kemb = K.coerce_embedding()
+            if Kemb is None:
+                raise TypeError("No embedding available for %s"%K)
+            Kemb = Kemb.codomain()
+            while Kemb.coerce_embedding() is not None:
+                Kemb = Kemb.coerce_embedding().codomain()
+            Lemb = L.coerce_embedding()
+            if Lemb is None:
+                raise TypeError("No embedding available for %s"%L)
+            Lemb = Lemb.codomain()
+            while Lemb.coerce_embedding() is not None:
+                Lemb = Lemb.coerce_embedding().codomain()
+                ambient_field = pushout(Kemb, Lemb)
+        else:
+            default_ambient_field = None
         gen_image = matching_root(K.polynomial().change_ring(L), K.gen(), ambient_field=ambient_field, margin=2)
+        if gen_image is None and default_ambient_field is not None:
+            ambient_field = default_ambient_field
+            gen_image = matching_root(K.polynomial().change_ring(L), K.gen(),
+                                      ambient_field=ambient_field, margin=2)
         if gen_image is None:
             raise ValueError, "No consistent embedding of all of %s into %s." % (K, L)
+
         NumberFieldEmbedding.__init__(self, K, L, gen_image)
         self.ambient_field = ambient_field