 Solution

Suppose first that $x$ is a purely inseparable algebraic element over $K$, sofor some $N\ge 0$. Then $m_x$ divides $T^{p^N}-y$. Over an algebraic closure the Frobenius endomorphism is injective, so this polynomial has only one distinct root:

Repeatedly use the zero-derivative criterion to writewith $q^{\prime }\ne 0$. The polynomial $q$ is irreducible and therefore separable by part (a), but all its roots are powers of the single root $x$. Hence $q$ has degree one. Since $m_x$ is monic,for some $a\in K$.

Conversely, if the minimal polynomial has this form, thenso $x$ is purely inseparable. Thus the minimal polynomial of a purely inseparable element is exactly