It is well-known that, in a real closed field $K$, every polynomial of degree>2 is reducible in $K$. But in this case the characteristic of $K$ is zero.

My question is: there exists a field $F$ of characteristic $p>0$ not perfect and a positive integer $n=n(F)$ such that all polynomials of degree$>n$ are reducible in $F$.

Edit: polynomials of degree prime with $p$