I was reading a paper on Function Fields. There is a statement like this. " Let $F \in F_q[t][x]$ here $x$ is a variable over $F_q[t]$and $deg_{x} F>0$ and $q$ is a power of $p$. Here author makes an assumption these $F$ are irreducible in $x$ and separable in $t$.
Now if I take $F = f(t)+x g(t)$, $f(t), g(t) \in F_q[t]$ are relatively prime polynomials this is a linear function in $x$ hence irreducible.
But if I take $F= f(t)+x^{2}g(t)$ what can be concluded? here $f(t) and $g(t)$ are relatively prime polynomials.
does it factorize as $(\sqrt{f(t)}+ix\sqrt{g(t)})(\sqrt{f(t)}-ix\sqrt{g(t)})$ in $F_q[t]$.
Any one can please check the correctness of this. Thank you