Suppose K is a field with characteristic p which is not a perfect field.
Then how do we prove that there does exist an irreducible polynomial which is not separable.
I have no idea how to proceed for this.what all i know is if K is a finite field then every irreducible polynomial is separable.
any hint/suggestion would be appreciated 🙂
Best Answer
As mentioned by Gerry Myerson let $K=F_p(t)$ and the sub-field $k=F(t^p)$ and finaly let the polynomial $$P=x^p-t^p\in K[x]$$ so in $K[x]$ we have $P=(x-t)^p$ and since $t^i\not\in k,\ i=1,\ldots,p-1$ then $P$ is irreductible in $k[x]$ but it's not separable as $t$ is its root with multiplicity $p$ in $K$.