# field, characteristic, surjective

• Mar 14th 2009, 06:29 PM
sophia782
field, characteristic, surjective
Suppose $\displaystyle F$ is a finite field of characteristic 3 and $\displaystyle \phi : F \rightarrow F$ is the map defined by $\displaystyle \phi(a)=a^3$. Prove that $\displaystyle \phi$ 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.
• Mar 14th 2009, 07:56 PM
ThePerfectHacker
Quote:

Originally Posted by sophia782
Suppose $\displaystyle F$ is a finite field of characteristic 3 and $\displaystyle \phi : F \rightarrow F$ is the map defined by $\displaystyle \phi(a)=a^3$. Prove that $\displaystyle \phi$ 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 $\displaystyle |F|$ is finite and $\displaystyle \phi: F\to F$ is injective it immeadiately means $\displaystyle \phi$ has to be surjective by the pigeonhole principle.