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Math Help - Generalizing Euler's Criterion

  1. #1
    Ant
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    Generalizing Euler's Criterion

    Euler's criterion - Wikipedia, the free encyclopedia

    The way I think about Euler's Criterion is that the order of the group Z cross p is phi(p)=p-1 so we know that  a^{(p-1)} \equiv 1 (mod p). Then also the only numbers which square to 1 are 1 and -1 so  a^{\frac{(p-1)}{2}} = 1,-1 (mod p)

    My question is can we generalize this to different group?

    Can we say:

     a^{\frac{\phi (m)}{2}} \equiv 1 or -1 (mod m)

    Thanks!
    Last edited by Ant; April 5th 2012 at 04:46 PM.
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  2. #2
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    Re: Generalizing Euler's Criterion

    Since \varphi(m) is an even number for all m>2 I think that answer is yes..
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  3. #3
    Ant
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    Re: Generalizing Euler's Criterion

    Ah yes, because  \phi(m) always includes a  p-1 term, which of course if even.
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