Prove that if $F$ is a finite field having q elements, then $a^{q-2}=a^{-1}$ for a nonzero $a \in F$

Question

This is a proposition from J.S.Golan's "The Linear Algebra a beginning graduate student ought to know".

I can't understand why it's so. Obviously, the order of $a$ is finite - higher than $1$ and lower than $q+1$. But what next?

Answer 1

If we can apply some group theory: there are $q-1$ units in the field (all non-zero elements), and these units form a group under multiplication. Applying Lagrange's theorem, we see that $a^{q-1} = 1$ for every $a \neq 0$. Now, $a^{q-2} = a^{q-1-1}$, so...