field homomorphism
Let and be fields.
Definition.
A field homomorphism is a function such that:
-
1.
for all
-
2.
for all
-
3.
If is injective and surjective, then we say that is a field isomorphism.
Lemma.
Let be a field homomorphism. Then is injective.
Proof.
Indeed, if is a field homomorphism, in particular it is a ring homomorphism. Note that the kernel of a ring homomorphism is an ideal and a field only has two ideals, namely . Moreover, by the definition of field homomorphism, , hence is not in the kernel of the map, so the kernel must be equal to . ∎
Remark: For this reason the terms “field homomorphism” and “field monomorphism” are synonymous. Also note that if is a field monomorphism, then
so there is a “copy” of in . In other words, if
is a field homomorphism then there exist a subfield of such that . Conversely, suppose there exists with isomorphic to . Then there is an isomorphism
and we also have the inclusion homomorphism
Thus the composition
is a field homomorphism.
Remark: Let be a field homomorphism. We claim that the characteristic of and must be the same. Indeed, since and then for all natural numbers . If the characteristic of is then in , and so the characteristic of is also . If the characteristic of is , then the characteristic of must be as well. For if in then , and since is injective by the lemma, we would have in as well.
Title | field homomorphism |
---|---|
Canonical name | FieldHomomorphism |
Date of creation | 2013-03-22 13:54:54 |
Last modified on | 2013-03-22 13:54:54 |
Owner | alozano (2414) |
Last modified by | alozano (2414) |
Numerical id | 9 |
Author | alozano (2414) |
Entry type | Definition |
Classification | msc 12E99 |
Synonym | field monomorphism |
Related topic | RingHomomorphism |
Defines | field homomorphism |
Defines | field isomorphism |