How can I see that $4$ is not a quartic residue?

Question

How can I see that $4$ is no quartic residue, i.e. there is no $t$ such that $t^4 \equiv 4 \mod p$ when $p\equiv 5 \mod 8$?

Answer 1

You are lucky that $4$ is a square.

You want to show that neither of the congruences $t^2 \equiv 2 \mod p$, $\ \ t^2 \equiv -2 \mod p$ has any solution.

Calculate the Legendre symbols (http://en.wikipedia.org/wiki/Legendre_symbol): \begin{eqnarray*} \left( \frac{2}{p}\right ) &=& (-1)^{\frac{p^2-1}{8}} = -1 \\ \left( \frac{-2}{p}\right ) &=& \left( \frac{-1}{p}\right )\cdot \left( \frac{2}{p}\right )= (-1)^{\frac{p-1}{2}} \cdot (-1) = 1 \cdot (-1) = -1 \end{eqnarray*}