I forgot why the next statement is true and it's bugging me endlessly...(adsbygoogle = window.adsbygoogle || []).push({});

If p is prime such that p =1 mod 4 then (-1) = x^2 mod p.

Now in Ashe's Algebriac Number theory notes (book?), he says that

[tex]((\frac{p-1}{2})!)^2= -1 mod p[/tex]

I am quite stumped as how to show this, he argues we just need to look at:

1*2*....((p-1)/2)*(-1)*(-2)....*(-(p-1)/2)

which on the one hand because p =1 mod 4 it equals [tex]((\frac{p-1}{2})!)^2 [/tex]

on the other hand it also equals (p-1)!, but why does this equal -1?

Thanks.

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# A question on square residue.

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