Define the Legendre symbol $\left(\frac{a}{p}\right)$.

State Gauss' lemma and use it to compute $\left(\frac{2}{p}\right)$ where $p$ is an odd prime.

Show that if $m \geqslant 4$ is a power of 2 , and $p$ is a prime dividing $2^{m}+1$, then $p \equiv 1(\bmod 4 m)$.