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Math Help - Fields, odd prime

  1. #1
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    Fields, odd prime

    Suppose F \subseteq K are fields, and that [K:F]=p, where p is an odd prime. Let \alpha \in K-F. Prove that K=F(\alpha^n) for any n, 1 \leq n <p.

    This one seems simple, I am thinking to prove first that the basis for F(\alpha) is 1, \alpha, \ldots, \alpha^{p-1} and then this would mean that F(\alpha)=F(\alpha^m) for any n, 1 \leq n <p. Is this a good way to go or is there an easier way I am not seeing now?
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    Quote Originally Posted by xboxlive89128 View Post
    Suppose F \subseteq K are fields, and that [K:F]=p, where p is an odd prime. Let \alpha \in K-F. Prove that K=F(\alpha^n) for any n, 1 \leq n <p.

    This one seems simple, I am thinking to prove first that the basis for F(\alpha) is 1, \alpha, \ldots, \alpha^{p-1} and then this would mean that F(\alpha)=F(\alpha^m) for any n, 1 \leq n <p. Is this a good way to go or is there an easier way I am not seeing now?
    Hint: p=[K:F]=[K:F(\alpha^n)][F(\alpha^n): F].
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