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Math Help - 2 Quadratic Reciprocity Proofs

  1. #1
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    Quadratic Reciprocity Proof (working now)

    Hi all, messed up using Latex at first. Okay, got two problems on quadratic reciprocity. I've worked up something of a proof for each, but I feel very uncertain. Thought maybe someone could give me some pointers.

    Okay, simple one first:
    Suppose p is a prime such that p\equiv 3 (mod 4), and let a be a QR modulo p.
    Show that x=a^\frac{p+1}{4} is a solution to
    x^2\equiv a(mod\, p)

    My proof:
    a - QR mod p \Rightarrow the Legendre symbol
    (\frac{a}{p})= 1 \equiv a^{\frac{p-1}{2}},
    so clearly 1\equiv a^{\frac{p-1}{2}} (mod\, p).
    Multiplying both sides by a yields
    a\equiv a^{\frac{p-1}{2}}*a\equiv a^{\frac{p+1}{2}}.
    But if x=a^{\frac{p+1}{4}},
    we have that x^{2}=a^{\frac{p+1}{2}}.
    We know that a^{\frac{p+1}{2}} is congruent to a modulo p, so
    x=a^{\frac{p+1}{4}} is a solution to the congruence.

    My trouble is that this seems too easy, and not once did I apply p\equiv 3(mod\, 4).

    Any pointers? Thanks in advance.
    Last edited by swashbucklord; June 19th 2011 at 04:29 PM. Reason: Used latex wrong
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  2. #2
    MHF Contributor chisigma's Avatar
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    Re: Quadratic Reciprocity Proof (working now)

    The 'solution' x= a^{\frac{p+1}{4}} must of course be an integer and that happens only if  p \equiv 3\ (\text{mod}\ 4)...

    Kind regards

    \chi \sigma
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