Separability of irreducible polynomial of degree 2

Question

Suppose $F$ is a field with $char(F)=2$. Let $f(x)$ be an irreducible polynomial in $F[x]$ with degree 2. Is it necessarily true that $f(x)$ is separable (that is, $f(x)$ does not have multiple roots in its splitting field)? I managed to prove that if $F$ is a finite field, then $f(x)$ cannot have multiple roots. I'm guessing we should look at an infinite field, but i can't come up with a proof or a counterexample.

Answer 1

Let $K=\Bbb F_2(T)$ the field of rational functions in $T$ over $\Bbb F_2$, and let $F=\Bbb F_2(T^2)$ be a subfield of $K$. Then $X^2-T^2$ is irreducible over $F$, but factors as $(X-T)^2$ over $K$, so it is inseparable.

This is a standard example, and appears frequently in textbooks.