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 $\displaystyle a^{(p-1)} \equiv 1 (mod p)$. Then also the only numbers which square to 1 are 1 and -1 so $\displaystyle a^{\frac{(p-1)}{2}} = 1,-1 (mod p) $

My question is can we generalize this to different group?

Can we say:

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

Thanks!

Re: Generalizing Euler's Criterion

Since $\displaystyle \varphi(m)$ is an even number for all $\displaystyle m>2$ I think that answer is yes..

Re: Generalizing Euler's Criterion

Ah yes, because $\displaystyle \phi(m) $ always includes a $\displaystyle p-1 $ term, which of course if even.