I was doing some algebra today and I came across the term 'purely inseparable extension'. However, I came across two different definitions of this term. We consider algebraic extensions and write $p$ = char($K$). The definitions go as follows:
A field extension $E/K$ is purely inseparable if and only if for every $x \in E\setminus K$ the minimal polynomial of $x$ over $K$ is not separable.
A field extension $E/K$ is purely inseparable if and only if for every $x \in E\setminus K$ there exists an $n$ such that $x^{p^n} \in K$.
To help convince myself that these different difinitions make sense, I tried to proof the equivalence between them. So I tried to prove the following:
For every $x \in E\setminus K$ the minimal polynomial of $x$ over $K$ is not separable if and only if for every $x \in E\setminus K$ there exists an $n$ such that $x^{p^n} \in K$.
However, I have not been succesfull in this endeavour. Can someone help me with this please? Thanks in advance.