# Math Help - field, characteristic, surjective

1. ## field, characteristic, surjective

Suppose $F$ is a finite field of characteristic 3 and $\phi : F \rightarrow F$ is the map defined by $\phi(a)=a^3$. Prove that $\phi$ is surjective.

How do I do this? I know how to prove that it is an injective ring homomorphism. But I don't know how to prove it is surjective. Thanks.

2. Originally Posted by sophia782
Suppose $F$ is a finite field of characteristic 3 and $\phi : F \rightarrow F$ is the map defined by $\phi(a)=a^3$. Prove that $\phi$ is surjective.

How do I do this? I know how to prove that it is an injective ring homomorphism. But I don't know how to prove it is surjective. Thanks.
Since $|F|$ is finite and $\phi: F\to F$ is injective it immeadiately means $\phi$ has to be surjective by the pigeonhole principle.