Let and be fields.
Let be a field homomorphism. Then is injective.
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
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.
|Date of creation||2013-03-22 13:54:54|
|Last modified on||2013-03-22 13:54:54|
|Last modified by||alozano (2414)|