## Over an imperfect field of characteristic p, there exist irreducible inseparable polynomials

Let $K$ be an imperfect field of characteristic $p$. Prove that there exist irreducible inseparable polynomials over $K$. Conclude that there exist inseparable finite extensions of $K$.

Let $\alpha \in K$ be an element which is not a $p$th power in $K$, and let $\zeta$ be a root of $x^p-\alpha$. In particular, $\zeta \notin K$. Now consider $q(x) = (x-\zeta)^p = x^p-\alpha$. Suppose $t(x)$ is an irreducible factor of $q$ over $K$. Now $t$ cannot be linear, since $\zeta \notin K$. But any factor of $q$ of degree at least 2 has a multiple root (since all the roots are $\zeta$). So $t$ is an irreducible inseparable polynomial over $K$.

Then $K(\zeta)$ is an inseparable extension of $K$.