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Math Help - Euler's criterion

  1. #1
    Junior Member
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    Euler's criterion

    "Let p = a prime. Show x^2 ≡ a (mod  p^2 ) has 0 solutions if x^2 ≡ a (mod p) has 0 solutions, or 2 solutions if x^2 ≡ a (mod p) has 2."

    I thought I should use Euler's criterion for this(?), which says x^2  a (mod p^2) if and only if a^{(p-1)/2} ≡ 1 (mod p), but I don't know where to go from there.

    Also, I know that a (mod p) can have only either exactly two solutions or zero solutions (by Lagrange), so the question remains how to map a (mod p) to a (mod  p^2 )

    Thanks !
    Last edited by kimberu; November 5th 2010 at 01:19 AM. Reason: added more info
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