# Using Euler theorem show that $a^{\frac{\varphi(m)}{2}}\equiv \pm1 \pmod m,~where~(a,m)=1$.

Euler Theorem: $a^{\varphi(m)}\equiv 1 \pmod m ,$ For $(a,m)=1.$

Using the above show that for $m=p^\alpha$ where $p$ is prime and $m\geq3$ $$a^{\frac{\varphi(m)}{2}}\equiv \pm1 \pmod m,~where~(a,m)=1$$ I know $\varphi(m)$ is even. So $\frac{\varphi(m)}{2}$ is some integer.

What do you get if you square $a^{\phi(m)/2}$?
• The square is $a^{\phi(m)}$ – user194772 Dec 6 '14 at 16:08
• First, what is $a^{\phi(m)}$ equal to? – Empy2 Dec 6 '14 at 16:13
• I think you are saying that since $$a^{\varphi(m)}\equiv1(mod~m)$$ then the square root of $a^{\varphi(m)}$ is $\equiv\pm1(mod~m)$? – user194772 Dec 6 '14 at 16:21