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I came across this problem and I believe Lagrange's theorem is the key to its solution. The question is:

Let $p$ be an odd prime. Prove that there is some integer $x$ such that $x^2 \equiv −1 \pmod p$ if and only if $p \equiv 1 \pmod 4$.

I appreciate any help. Thanks.

  • 0
    If you mean [Lagrange's four-square theorem](http://en.wikipedia.org/wiki/Lagrange%27s_four-square_theorem), then it is probably overkill if it works at all. See [Fermat's theorem on sums of two squares](http://en.wikipedia.org/wiki/Fermat%27s_theorem_on_sums_of_two_squares). which points to several proofs.2012-03-19
  • 0
    Read up on the Legendre Symbol, which uses Euler's criterion, which can be proved with Lagrange's Theorem.2012-03-19
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    @ lhf I meant Lagrange's theorem in group theory. http://en.wikipedia.org/wiki/Lagrange's_theorem_(group_theory)2012-03-19
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    For the simpler theorem (which is equivalent to the sum of two squares theorem), you can use [Wilson's theorem](http://en.wikipedia.org/wiki/Wilson%27s_theorem#Applications).2012-03-19

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