Show that if p is an odd prime, with p $\equiv 3 \pmod 4$ then $(\mathbb{Z}^*_{p})^4$ = $(\mathbb{Z}^*_{p})^2$.

Question

Show that if p is an odd prime, with p $\equiv 3 \pmod 4$, then $(\mathbb{Z}^*_{p})^4$ = $(\mathbb{Z}^*_{p})^2$.

Any help is appreciated.

Answer 1

Hint: Assume that $m\equiv a^2\not\equiv0$. We were given that $p-1=4\ell+2$. So by Little Fermat $$ m\equiv a^2\equiv a^2a^{p-1}\equiv a^{4(\ell+1)}. $$