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March 14th, 2009, 06:35 PM   #1
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field, characteristic, surjective

Suppose is a finite field of characteristic 3 and is the map defined by . Prove that 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.
sophia782 is offline  
March 14th, 2009, 09:15 PM   #2
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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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characteristic, field, surjective

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