Generalizing Euler's Criterion

**Ant** · April 5th 2012, 03:42 PM

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!