reviewer comments

src/sage/rings/finite_rings/finite_field_base.pyx

@@ -1948,6 +1948,8 @@ cdef class FiniteField(Field):
               To:   Algebraic closure of Finite Field of size 2
               Defn: a |--> ...
         """
+        if self.characteristic() != K.characteristic():
+            raise ValueError(f'no embedding from {self} to {K}: incompatible characteristics')
         try:
             return super().an_embedding(K)
         except (NotImplementedError, ValueError):
@@ -1985,42 +1987,9 @@ cdef class FiniteField(Field):
                From: Finite Field in z2 of size 2^2
                To:   Algebraic closure of Finite Field of size 2
                Defn: z2 |--> z2 + 1]
-
-        ::
-
-            sage: CyclotomicField(5).embeddings(QQbar)
-            [
-            Ring morphism:
-              From: Cyclotomic Field of order 5 and degree 4
-              To:   Algebraic Field
-              Defn: zeta5 |--> 0.3090169943749474? + 0.9510565162951536?*I,
-            Ring morphism:
-              From: Cyclotomic Field of order 5 and degree 4
-              To:   Algebraic Field
-              Defn: zeta5 |--> -0.8090169943749474? + 0.5877852522924731?*I,
-            Ring morphism:
-              From: Cyclotomic Field of order 5 and degree 4
-              To:   Algebraic Field
-              Defn: zeta5 |--> -0.8090169943749474? - 0.5877852522924731?*I,
-            Ring morphism:
-              From: Cyclotomic Field of order 5 and degree 4
-              To:   Algebraic Field
-              Defn: zeta5 |--> 0.3090169943749474? - 0.9510565162951536?*I
-            ]
-            sage: CyclotomicField(3).embeddings(CyclotomicField(7))
-            [ ]
-            sage: CyclotomicField(3).embeddings(CyclotomicField(6))
-            [
-            Ring morphism:
-              From: Cyclotomic Field of order 3 and degree 2
-              To:   Cyclotomic Field of order 6 and degree 2
-              Defn: zeta3 |--> zeta6 - 1,
-            Ring morphism:
-              From: Cyclotomic Field of order 3 and degree 2
-              To:   Cyclotomic Field of order 6 and degree 2
-              Defn: zeta3 |--> -zeta6
-            ]
         """
+        if self.characteristic() != K.characteristic():
+            return []
         if K not in FiniteFields():
             from sage.rings.algebraic_closure_finite_field import AlgebraicClosureFiniteField_generic
             if not isinstance(K, AlgebraicClosureFiniteField_generic):

src/sage/rings/number_field/number_field.py

@@ -9345,6 +9345,41 @@ def embeddings(self, K):
               Defn: a |--> 1.25992104989487
             ]
 
+        Some more (possible and impossible) embeddings of cyclotomic fields::
+
+            sage: CyclotomicField(5).embeddings(QQbar)
+            [
+            Ring morphism:
+              From: Cyclotomic Field of order 5 and degree 4
+              To:   Algebraic Field
+              Defn: zeta5 |--> 0.3090169943749474? + 0.9510565162951536?*I,
+            Ring morphism:
+              From: Cyclotomic Field of order 5 and degree 4
+              To:   Algebraic Field
+              Defn: zeta5 |--> -0.8090169943749474? + 0.5877852522924731?*I,
+            Ring morphism:
+              From: Cyclotomic Field of order 5 and degree 4
+              To:   Algebraic Field
+              Defn: zeta5 |--> -0.8090169943749474? - 0.5877852522924731?*I,
+            Ring morphism:
+              From: Cyclotomic Field of order 5 and degree 4
+              To:   Algebraic Field
+              Defn: zeta5 |--> 0.3090169943749474? - 0.9510565162951536?*I
+            ]
+            sage: CyclotomicField(3).embeddings(CyclotomicField(7))
+            [ ]
+            sage: CyclotomicField(3).embeddings(CyclotomicField(6))
+            [
+            Ring morphism:
+              From: Cyclotomic Field of order 3 and degree 2
+              To:   Cyclotomic Field of order 6 and degree 2
+              Defn: zeta3 |--> zeta6 - 1,
+            Ring morphism:
+              From: Cyclotomic Field of order 3 and degree 2
+              To:   Cyclotomic Field of order 6 and degree 2
+              Defn: zeta3 |--> -zeta6
+            ]
+
         Test that :issue:`15053` is fixed::
 
             sage: K = NumberField(x^3 - 2, 'a')

src/sage/rings/number_field/number_field_rel.py

@@ -2072,6 +2072,9 @@ def embeddings(self, K):
             sage: f[0](a+b)
             -0.62996052494743693 - 0.091123635971721295*I
         """
+        if K.characteristic():
+            return Sequence([], immutable=True, check=False, universe=self.Hom(K))
+
         try:
             # this should be concordant with automorphisms
             return self.__embeddings[K]

src/sage/rings/rational_field.py

@@ -601,7 +601,8 @@ def signature(self):
 
     def embeddings(self, K):
         r"""
-        Return list of the one embedding of `\QQ` into `K`, if it exists.
+        Return the list containing the unique embedding of `\QQ` into `K`,
+        if it exists, and an empty list otherwise.
 
         EXAMPLES::
 
@@ -612,16 +613,17 @@ def embeddings(self, K):
                From: Rational Field
                To:   Cyclotomic Field of order 5 and degree 4]
 
-        `K` must have characteristic 0::
+        The field `K` must have characteristic `0` for an embedding of `\QQ`
+        to exist::
 
             sage: QQ.embeddings(GF(3))
-            Traceback (most recent call last):
-            ...
-            ValueError: no embeddings of the rational field into K.
+            []
         """
-        if K.characteristic():
-            raise ValueError("no embeddings of the rational field into K.")
-        return [self.hom(K)]
+        if K.characteristic() == 0:
+            v = [self.hom(K)]
+        else:
+            v = []
+        return Sequence(v, check=False, universe=self.Hom(K))
 
     def automorphisms(self):
         r"""

src/sage/rings/ring.pyx

@@ -1526,6 +1526,9 @@ cdef class Field(CommutativeRing):
               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()