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Math Help - P-adic Number help please

  1. #1
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    P-adic Number help please

    Hi

    I was wondering if anyone could help me on this problem I am having.

    I have the equation:

    y^2 = 2( x^4 -17)
    (which I am showing has no real solutions but has solutions in the p-adic numbers i.e. a counterexample to the hasse principle)

    I need to show that it has a non trivial solution in R and in Q_p (p-adic numbers for p=2, 17 (the 2-adics and 17-adic)

    Any help or advice would be so much apprciated! I`m completely stumped!

    THANKS!!

    Cara

    xxxx
    Last edited by mr fantastic; May 10th 2010 at 06:18 PM.
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  2. #2
    MHF Contributor chiph588@'s Avatar
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    Champaign, Illinois
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    What do you mean by no trivial solution in  \mathbb{R} ?

    In  \mathbb{R} we have solutions  \left(x,\pm\sqrt{2(x^4-17)}\right) for all  |x|>\sqrt[4]{17} .
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  3. #3
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    Sorry that was a typo! Thats meant to be in Q.
    this equation has nontrivial solutions over
    Qp for all prime numbers p and over R but has no nontrivial solutions over Q.
    I need to show that although there is no solution in Q there is a solution in Qp . But I don`t know how to show that for the case where modulo p=2 and modilo p=17?

    Its a special case, but its completely confused me.
    Thanks so much for the reply!

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