In a finite field product of non-square elements is a square I came across one problem in a finite field as follows:

Let $F$ be a finite field. Show that if $a, b\in F$ both are non-squares, then $ab$ is a square. 

I wanted to prove it by using the idea of Biquadratic field extension. But there is no biquadratic extension over finite fields. Please, any hints for proving above fact? Thanks. 

Hint: I assume, $a,b,\neq 0$ otherwise it is obvious. $F^*$ is cyclic. Suppose it is generated by $x$, you can write $a=x^n, b=x^m$, $n,m$ odd. Then $n+m$ is even.
