Modular Arithmetic, roots of integers?

Question

Apologies that the title is vague, but I’m not quite sure where this question would fall under modular arithmetic. The question is:

Let $k$ be a natural number with hcf$(k,p-1)=1$ where $p$ is prime. Prove that every integer has a $k^{th}$ root modulo p.

I really don’t even know where to begin with this so could someone please give me an indication of how to start the proof and then hopefully from there I should be able to finish the problem.

Answer 1

There exists $r,s \in \Bbb Z$ such that $kr + (p-1)s = 1$. By Fermat's Little Theorem,

$$a \equiv a^{kr + (p-1)s} \equiv (a^r)^k \pmod p.$$