Leq $\mathbb{F}_q$ be a finite field of characteristic $2$. Let $g(x)$ and $f(x)\in\mathbb{F}_q[x]$ be such that $g(x)$ is irreducible and ${g^{\prime}(x)}^2 f(x) \equiv {f^{\prime}(x)}^2 \mod{g(x)}$.
Consider the polynomial $F(x,y)=y^2 + g(x) y + f(x)$. From the algebraic-geometric point of view, $F(x,y)=0$ defines a singular hyperelliptic curve $\mathbb{F}_q$.
Is it true that $F(x,y)$ is reducible over $\mathbb{F}_q$? over $\overline{\mathbb{F}_q}$?
