proving that -1 is square if p = 1 mod 4

Asked Nov 02 '15 at 05:11

Active Nov 02 '15 at 05:11

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I have to find $gcd(x^{P-1}-1,x^2+1)$ in $\Bbb{Z}/p\Bbb{Z}[X]$ for some prime number p in order to show that $-1$ is a square $\pmod p$ iff $p\equiv1\pmod4$. I really have no idea where to start. I don't see how to get the gcd given that p is not a number. Any help would be appreciated.

Thanks

asked Nov 02 '15 at 05:11

Rob

Try calculating $\gcd(x^{2n} - 1, x^2 + 1)$ (actually working through the calculations) for $n=1,2,3,4$ and see if you can observe and understand a pattern there. – Erick Wong Nov 02 '15 at 05:22
A very similar question, asking for the use of the same polynomial tools, was asked yesterday. – André Nicolas Nov 02 '15 at 06:39
I think the duplicate candidate suggested by André is a better match here, so I used it. – Jyrki Lahtonen Nov 03 '15 at 14:24

proving that -1 is square if p = 1 mod 4

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