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Math Help - finite field (and subfields thereof)

  1. #1
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    finite field (and subfields thereof)

    K is a finite field, F is a subfield of K, and m is a positive integer.

    L=\{a \in K \vert a^{p^m} \in F \}.

    Show that L is a subfield of K containing F. Moreover, show that L=F.

    Easy enough to show L is a subfield of K, and I have a proof that L=F assuming that L contains F, but I do not know why L must contain F.

    Thanks in advance (and afterwards as well, naturally).
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  2. #2
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    Quote Originally Posted by mylestone View Post
    K is a finite field, F is a subfield of K, and m is a positive integer.

    L=\{a \in K \vert a^{p^m} \in F \}.

    Show that L is a subfield of K containing F. Moreover, show that L=F.

    Easy enough to show L is a subfield of K, and I have a proof that L=F assuming that L contains F, but I do not know why L must contain F.
    you're kidding, right? well, if a \in F \subseteq K, then obviously a^{p^m} will also be in F (because F is closed under multiplication). thus F \subseteq L.
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  3. #3
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    Quote Originally Posted by NonCommAlg View Post
    you're kidding, right? well, if a \in F \subseteq K, then obviously a^{p^m} will also be in F (because F is closed under multiplication). thus F \subseteq L.
    Jeepers ...apologies, and thank you.
    Last edited by mr fantastic; July 10th 2011 at 04:29 PM.
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