Hi,

By a primitive 4th root of unity, I assume you mean a y such that y^{4}is 1 mod p, but no smaller power is 1. Now if you know about cyclic groups, your problem is easy. However, even without this here's a proof.

There is a primitive 4th root of unity mod p if and only if 4 divides p-1.

Proof of only if:

Let x be a primitive root mod p; i.e. every non-zero element of Z_{p}is a power of x. (You need to know such exist.) So suppose y=x^{m}is a primitive 4th root. I need the following fact: the order of x^{m}mod p is (p-1)/gcd(m,p-1). (Order is the least positive power k with (x^{m})^{k}= 1 mod p; this is a standard result from cyclic groups. But it's not hard to prove separately.) So p-1=4gcd(m,p-1) or 4 divides p-1. QED.