Prove-If n is prime $\mathbb{Z}_n$ is a field. I need to prove that $\mathbb{Z}_n$ is a field if and only if $n$ is prime. And I proved the forward. 
But I am not sure how to prove the backward, 'if n is prime $\mathbb{Z}_n$ is a field.
'
What can I assume from the assumption $n$ is prime? 
and What do I need to show? 

Other than showing trivial axioms, you need to show that every element in $Z/pZ$ has a multiplicative inverse. In other words, given $k$, you need to find $m$ such that $k \cdot m \equiv 1 \pmod p$
This, though, is equivalent to showing that there exists $n$ such that $km = pn + 1$. Because $p$ is a prime, we know that $\gcd(k, p) = 1$. By Euclid's Algorithm/Linear Diophantine Equations, there must exist $(m_0 , n_0)$ such that $km_0 +pn_0 = 1$. The result follows.
I can clarify any of this if you'd like - feel free to comment.
Cheers!
