Prove $1+a^{p-1}+...a^{(p-1)^2}$ is divisible by prime p,given a and p are co-prime.

Question

Prove $1+a^{p-1}+...a^{(p-1)^2}$ is divisible by p,given a and p are co-prime.

Answer 1

Note that $a^{k(p-1)} \equiv 1 \pmod{p}$ by Fermat's Little Theorem, so $a^{(p-1)^2}+a^{(p-2)(p-1)}+\ldots+a^{p-1}+1\equiv 1+1+\ldots+1 \equiv p \equiv 0 \pmod{p}$. Thus, $p \mid a^{(p-1)^2}+a^{(p-2)(p-1)}+\ldots+a^{p-1}+1$.