Prove that for a nonzero element $a$ of a field $F$ with $q$ elements $a^{-1} = a^{q-2}$

Question

This is a statement from "The Linear Algebra a Beginning Graduate Student Ought to Know" by Jonathan S. Golan.

Let $F$ be a field having $q$ elements and let $a$ be it's nonzero element. Then $a^{-1} = a^{q-2}$

If $a = 1$, than it holds. If $a \neq 1$, then $a^{q-1} = 1$( the multiplicative order $|a|$ of a divides $q-1$ ).

What can be done next?

Answer 1

$a^{q-1} = 1 \Leftrightarrow a^{q-1}a^{-1} = a^{-1} \Leftrightarrow a^{q-2} = a^{-1}$