## The Set of Square Elements in the Multiplicative Group $(\Zmod{p})^*$

## Problem 616

Suppose that $p$ is a prime number greater than $3$.

Consider the multiplicative group $G=(\Zmod{p})^*$ of order $p-1$.

**(a)** Prove that the set of squares $S=\{x^2\mid x\in G\}$ is a subgroup of the multiplicative group $G$.

**(b)** Determine the index $[G : S]$.

**(c)** Assume that $-1\notin S$. Then prove that for each $a\in G$ we have either $a\in S$ or $-a\in S$.