Showing that $\mathbb{Z}_N$ is a field if $N$ is prime I know that $N$ being prime is a necessary and sufficient condition for $\mathbb{Z}_N$ to be a field. 

I know how to prove that it's necessary but I'm not sure how to prove that this is a sufficient condition on $N$.
Hint In this case, it's not too hard to check the field axioms directly. Alternately, one can show that $\langle p \rangle$ is a maximal ideal of $\Bbb Z$ for any prime $p$; in fact, this property is close to primeness.
Hint:  Prove  that any non-zero element from $\mathbb{Z}_n$  has multiplicative inverse.