Simplify the logic for finding coercions between embedded number fields

Author: mezzarobba

src/sage/rings/number_field/number_field.py

@@ -6139,35 +6139,18 @@ def _coerce_map_from_(self, R):
         from sage.rings.number_field.order import is_NumberFieldOrder
         if is_NumberFieldOrder(R) and self.has_coerce_map_from(R.number_field()):
             return self._generic_convert_map(R)
-        if is_NumberField(R) and R != QQ:
-            if R.coerce_embedding() is not None:
-                if self.coerce_embedding() is not None:
-                    try:
-                        try:
-                            return number_field_morphisms.EmbeddedNumberFieldMorphism(R, self)
-                        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
-                    # coercions, as was pointed out at ticket #8800
+        # R is not QQ by the above tests
+        if is_NumberField(R) and R.coerce_embedding() is not None:
+            if self.coerce_embedding() is not None:
+                try:
+                    return number_field_morphisms.EmbeddedNumberFieldMorphism(R, self)
+                except ValueError: # no common embedding found
                     return None
+            else:
+                # R is embedded, self isn't. So, we could only have
+                # the forgetful coercion. But this yields to non-commuting
+                # coercions, as was pointed out at ticket #8800
+                return None
 
     def base_field(self):
         """

src/sage/rings/number_field/number_field_morphisms.pyx

@@ -20,6 +20,7 @@ fields (generally `\RR` or `\CC`).
 #                  http://www.gnu.org/licenses/
 #*****************************************************************************
 
+import sage.rings.complex_double
 
 from sage.structure.element cimport Element
 from sage.categories.morphism cimport Morphism
@@ -173,34 +174,34 @@ cdef class EmbeddedNumberFieldMorphism(NumberFieldEmbedding):
 
         """
         if ambient_field is None:
-            from sage.rings.complex_double import CDF
-            default_ambient_field = CDF
-            Kemb = K.coerce_embedding()
-            if Kemb is None:
+            if K.coerce_embedding() is None:
                 raise TypeError("No embedding available for %s"%K)
-            Kemb = Kemb.codomain()
+            Kemb = K
             while Kemb.coerce_embedding() is not None:
                 Kemb = Kemb.coerce_embedding().codomain()
-            Lemb = L.coerce_embedding()
-            if Lemb is None:
+            if L.coerce_embedding() is None:
                 raise TypeError("No embedding available for %s"%L)
-            Lemb = Lemb.codomain()
+            Lemb = L
             while Lemb.coerce_embedding() is not None:
                 Lemb = Lemb.coerce_embedding().codomain()
-                ambient_field = pushout(Kemb, Lemb)
+            ambient_field = pushout(Kemb, Lemb)
+            candidate_ambient_fields = [ambient_field, sage.rings.complex_double.CDF]
+            try:
+                candidate_ambient_fields.append(ambient_field.algebraic_closure())
+            except NotImplementedError:
+                pass
+        else:
+            candidate_ambient_fields = [ambient_field]
+
+        for ambient_field in candidate_ambient_fields:
+            gen_image = matching_root(K.polynomial().change_ring(L), K.gen(), ambient_field=ambient_field, margin=2)
+            if gen_image is not None:
+                NumberFieldEmbedding.__init__(self, K, L, gen_image)
+                self.ambient_field = ambient_field
+                return
         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
-
     def section(self):
         """
         EXAMPLES::