# Math Help - Congruence in Z

1. ## Congruence in Z

Let p be prime in Z with $p \equiv 1 \ (mod \ 4).$ Show that there is an integer $n \in Z$ with $n^2 \equiv -1 \ (mod \ p)$

Proof. Now I have p - 1 = 4k for some integers k. Then p = 4k - 1, pk = 4k^2 - k, so I have a square of something but with an extra k behind, how can I continue?

Thank you.

2. Originally Posted by tttcomrader
Let p be prime in Z with $p \equiv 1 \ (mod \ 4).$ Show that there is an integer $n \in Z$ with $n^2 \equiv -1 \ (mod \ p)$
Use Euler's criterion. Thus, $-1$ is a quadradic residue if and only if $(-1)^{(p-1)/2} \equiv 1 (\bmod p)$ hence if $p\equiv 1(\bmod 4)$ then $(p-1)/2$ is even and the congruence holds.