# Generalizing Euler's Criterion

• April 5th 2012, 03:42 PM
Ant
Generalizing Euler's Criterion
Euler's criterion - Wikipedia, the free encyclopedia

The way I think about Euler's Criterion is that the order of the group Z cross p is phi(p)=p-1 so we know that $a^{(p-1)} \equiv 1 (mod p)$. Then also the only numbers which square to 1 are 1 and -1 so $a^{\frac{(p-1)}{2}} = 1,-1 (mod p)$

My question is can we generalize this to different group?

Can we say:

$a^{\frac{\phi (m)}{2}} \equiv 1 or -1 (mod m)$

Thanks!
• April 6th 2012, 01:05 AM
princeps
Re: Generalizing Euler's Criterion
Since $\varphi(m)$ is an even number for all $m>2$ I think that answer is yes..
• April 6th 2012, 01:25 AM
Ant
Re: Generalizing Euler's Criterion
Ah yes, because $\phi(m)$ always includes a $p-1$ term, which of course if even.