For a prime $\displaystyle p$, let $\displaystyle \mathbb{Z}_{p}$ be the set of all equivalence classes of the relation defined by:

$\displaystyle aRb \Leftrightarrow a \equiv b(mod p)$.

Prove or disprove: $\displaystyle \mathbb{Z}_{p}$ is a field.

So I already showed that the set has addition and multiplication well-defined. The only thing I'm having trouble with is showing that for any $\displaystyle a \in \mathbb{Z}_{p}$ there exists $\displaystyle a^{-1} \in \mathbb{Z}_{p}$ such that $\displaystyle aa^{-1} = 1$.

How would I go about showing that exactly?