Euler's Criterion

Proposition (Euler's Criterion) Let be an odd prime and Then



    Proof. The only values for are So it suffices to consider the cases and
    If then has a solution say Then by Fermat's Little theorem,  
    


since Conversely, if then has no solution. The key idea is that we can group together the integers into pairs each with product of . Then multiplying these pairs together, and using Wilson's theorem, we have:



To see why we can do this, note that means has exactly one solution say and this must happen precisely when         

Example (Quadratic Character of -1) If is an odd prime, then



    Solution. Every odd prime is of the form (that is ) or of the form   (that is ). In the first case, Euler's Criterion yields



because is even. In the latter case,   is odd and so Euler's Criterion yields

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