also moving an_embedding to categories

Author: fchapoton

src/sage/categories/fields.py

@@ -277,6 +277,58 @@ def algebraic_closure(self):
             """
             raise NotImplementedError("algebraic closures of general fields not implemented")
 
+        def an_embedding(self, K):
+            r"""
+            Return some embedding of this field into another field `K`,
+            and raise a :class:`ValueError` if none exists.
+
+            EXAMPLES::
+
+                sage: GF(2).an_embedding(GF(4))
+                Ring morphism:
+                  From: Finite Field of size 2
+                  To:   Finite Field in z2 of size 2^2
+                  Defn: 1 |--> 1
+                sage: GF(4).an_embedding(GF(8))
+                Traceback (most recent call last):
+                ...
+                ValueError: no embedding from Finite Field in z2 of size 2^2 to Finite Field in z3 of size 2^3
+                sage: GF(4).an_embedding(GF(16))
+                Ring morphism:
+                  From: Finite Field in z2 of size 2^2
+                  To:   Finite Field in z4 of size 2^4
+                  Defn: z2 |--> z4^2 + z4
+
+            ::
+
+                sage: CyclotomicField(5).an_embedding(QQbar)
+                Coercion map:
+                  From: Cyclotomic Field of order 5 and degree 4
+                  To:   Algebraic Field
+                sage: CyclotomicField(3).an_embedding(CyclotomicField(7))
+                Traceback (most recent call last):
+                ...
+                ValueError: no embedding from Cyclotomic Field of order 3 and degree 2 to Cyclotomic Field of order 7 and degree 6
+                sage: CyclotomicField(3).an_embedding(CyclotomicField(6))
+                Generic morphism:
+                  From: Cyclotomic Field of order 3 and degree 2
+                  To:   Cyclotomic Field of order 6 and degree 2
+                  Defn: zeta3 -> zeta6 - 1
+            """
+            if self.characteristic() != K.characteristic():
+                raise ValueError(f'no embedding from {self} to {K}: incompatible characteristics')
+
+            H = self.Hom(K)
+            try:
+                return H.natural_map()
+            except TypeError:
+                pass
+            from sage.categories.sets_cat import EmptySetError
+            try:
+                return H.an_element()
+            except EmptySetError:
+                raise ValueError(f'no embedding from {self} to {K}')
+
         def prime_subfield(self):
             """
             Return the prime subfield of ``self``.
@@ -699,22 +751,6 @@ def vector_space(self, *args, **kwds):
             """
             return self.free_module(*args, **kwds)
 
-        def _pseudo_fraction_field(self):
-            """
-            The fraction field of ``self`` is always available as ``self``.
-
-            EXAMPLES::
-
-                sage: QQ._pseudo_fraction_field()
-                Rational Field
-                sage: K = GF(5)
-                sage: K._pseudo_fraction_field()
-                Finite Field of size 5
-                sage: K._pseudo_fraction_field() is K
-                True
-            """
-            return self
-
     class ElementMethods:
         def euclidean_degree(self):
             r"""

src/sage/rings/ring.pyx

@@ -704,58 +704,6 @@ cdef class Field(CommutativeRing):
     """
     _default_category = _Fields
 
-    def an_embedding(self, K):
-        r"""
-        Return some embedding of this field into another field `K`,
-        and raise a :class:`ValueError` if none exists.
-
-        EXAMPLES::
-
-            sage: GF(2).an_embedding(GF(4))
-            Ring morphism:
-              From: Finite Field of size 2
-              To:   Finite Field in z2 of size 2^2
-              Defn: 1 |--> 1
-            sage: GF(4).an_embedding(GF(8))
-            Traceback (most recent call last):
-            ...
-            ValueError: no embedding from Finite Field in z2 of size 2^2 to Finite Field in z3 of size 2^3
-            sage: GF(4).an_embedding(GF(16))
-            Ring morphism:
-              From: Finite Field in z2 of size 2^2
-              To:   Finite Field in z4 of size 2^4
-              Defn: z2 |--> z4^2 + z4
-
-        ::
-
-            sage: CyclotomicField(5).an_embedding(QQbar)
-            Coercion map:
-              From: Cyclotomic Field of order 5 and degree 4
-              To:   Algebraic Field
-            sage: CyclotomicField(3).an_embedding(CyclotomicField(7))
-            Traceback (most recent call last):
-            ...
-            ValueError: no embedding from Cyclotomic Field of order 3 and degree 2 to Cyclotomic Field of order 7 and degree 6
-            sage: CyclotomicField(3).an_embedding(CyclotomicField(6))
-            Generic morphism:
-              From: Cyclotomic Field of order 3 and degree 2
-              To:   Cyclotomic Field of order 6 and degree 2
-              Defn: zeta3 -> zeta6 - 1
-        """
-        if self.characteristic() != K.characteristic():
-            raise ValueError(f'no embedding from {self} to {K}: incompatible characteristics')
-
-        H = self.Hom(K)
-        try:
-            return H.natural_map()
-        except TypeError:
-            pass
-        from sage.categories.sets_cat import EmptySetError
-        try:
-            return H.an_element()
-        except EmptySetError:
-            raise ValueError(f'no embedding from {self} to {K}')
-
 
 cdef class Algebra(Ring):
     def __init__(self, base_ring, *args, **kwds):