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 March 14th, 2009, 06:35 PM #1 Newbie   Joined: Mar 2009 Posts: 6 Thanks: 0 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.
 March 14th, 2009, 09:15 PM #2 Member   Joined: Aug 2008 Posts: 84 Thanks: 0 Re: field, characteristic, surjective Remember that a function between two finite sets of the same cardinality is injective iff it is surjective. Injectivity, as you proved, is not difficult to come by, thus we get surjectivity for free! This is in fact a field automorphism.

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