Let $\mathbb{F}_q$ be finite field with characteristic 2 and consider the polynomial ring $A=\mathbb{F}_q[T]$. Denote $F=\mathbb{F}_q(T)$.
Let $f(x)=ax^2+bx+c\in A[x]$ be irreducible. I'm trying to prove that if $\deg(a)+\deg(c)$ is odd, then the splitting field $K$ of $f(x)$ is imaginary, i.e. $f(x)$ remains irreducible over the completion $F_\infty$. I've done a few examples, but I'm not really seeing a pattern that would help me prove this. Any hints?