Questions about eigenvalues of matrices in $\operatorname{GL}_2(\mathbb{F}_q)$.

Question

I have some questions about eigenvalues of matrices in $\operatorname{GL}_2(\mathbb{F}_q)$. Since $\mathbb{F}_q$ is not algebraically closed, it is possible that some $g \in \operatorname{GL}_2(\mathbb{F}_q)$ has eigenvalues which are not in $\mathbb{F}_q$. It is said that in this case, the eigenvalues of $g$ must be in $\mathbb{F}_q(\sqrt{\varepsilon})$, $\varepsilon \in \mathbb{F}_q \backslash \mathbb{F}_q^2$. Why this is true? Thank you very much.

Answer 1

Hint: You know that the eigenvalues satisfy the characteristic polynomial which is degree $2$. Apply the quadratic formula.

EDIT: As Jyrki points out, we have a bit of a sticking point in characteristic $2$ sinc the quadratic formula doesn't work there. But, we're still in a quadratic extension. In other words, the classic field theory problem "all quadratic extensions are obtained by adjoining a squareroot" usually follows with "except in characteristic $2$" :)