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 $p$