Irreducible polynomial of the form $x^4+ax^2+b \in K[x]$, where $K$ is a field with char$K \neq 2$, is separable?

Question

Let $K$ be a field with char$K \neq2$. My question is, if $x^4+ax^2+b \in K[x]$ is irreducible (over $K$), then it is true that this polynomial is separable? I of course know that if this is true when $K$ is finite, or when char$K=0$. Is there a counterexample?

Answer 1

The only way a quartic can be inseparable is over characteristic $2$ or $3$.

In characteristic $3$ an irreducible inseparable polynomial is a polynomial in $x^3$. The only way $x^4+ax^2+b$ can then be inseparable is if it factors into the form $(x-r)(x^3-s)$. But for that to have vanishing $x$ and $x^3$ coefficients, we need $r=s=0$ and so $a=b=0$. Of course $x^4$ is separable.