field homomorphism

Let F and K be fields.


A field homomorphism is a function ψ:FK such that:

  1. 1.

    ψ(a+b)=ψ(a)+ψ(b) for all a,bF

  2. 2.

    ψ(ab)=ψ(a)ψ(b) for all a,bF

  3. 3.


If ψ is injectivePlanetmathPlanetmath and surjectivePlanetmathPlanetmath, then we say that ψ is a field isomorphism.


Let ψ:FK be a field homomorphism. Then ψ is injective.


Indeed, if ψ is a field homomorphism, in particular it is a ring homomorphismMathworldPlanetmath. Note that the kernel of a ring homomorphism is an ideal and a field F only has two ideals, namely {0},F. Moreover, by the definition of field homomorphism, ψ(1)=1, hence 1 is not in the kernel of the map, so the kernel must be equal to {0}. ∎

Remark: For this reason the terms “field homomorphism” and “field monomorphism” are synonymous. Also note that if ψ is a field monomorphism, then


so there is a “copy” of F in K. In other words, if


is a field homomorphism then there exist a subfieldMathworldPlanetmath H of K such that HF. Conversely, suppose there exists HK with H isomorphicPlanetmathPlanetmathPlanetmath to F. Then there is an isomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmath


and we also have the inclusion homomorphismMathworldPlanetmathPlanetmathPlanetmath


Thus the composition


is a field homomorphism.

Remark: Let ψ:FK be a field homomorphism. We claim that the characteristicPlanetmathPlanetmath of F and K must be the same. Indeed, since ψ(1F)=1K and ψ(0F)=0K then ψ(n1F)=n1K for all natural numbersMathworldPlanetmath n. If the characteristic of F is p>0 then 0=ψ(p1)=p1 in K, and so the characteristic of K is also p. If the characteristic of F is 0, then the characteristic of K must be 0 as well. For if p1=0 in K then ψ(p1)=0, and since ψ is injective by the lemma, we would have p1=0 in F 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