Let $K$ be an infinite algebraic extension of a finite field and $a,b \in K$ are both in the same finite subfield of $K$...

Question

Let $K$ be an infinite algebraic extension of a finite field. If $a,b$ are any two elements of $K$ then they lie in a finite subfield of $K$. Then we know that if $a$ and $b$ are both nonsquares then $ab$ is a square.

Let Let $K^*$ denote the multiplicative group of units and $(K^*)^2 =\{a^2 : a \in K^* \}$. The bold part implies that $[K^*:(K^*)^2]= 1 \text{ or } 2$. Why is this also?

Can somebody explain to me the bold part? Thank you.