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Math Help - field, characteristic, surjective

  1. #1
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    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.
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  2. #2
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    Quote Originally Posted by sophia782 View Post
    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.
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