Assume that $K$ is a complete field under a discrete valuation with Dedekind ring $A$ and maximal ideal $\mathfrak p$ and $A\diagup\mathfrak p$ is perfect. Let $e$ be a positive integer not divisible by $E$. Let $E$ be a finite extension of $K$, $\pi_0$ a prime element in $\mathfrak p$, and $\beta$ an element of $E$ such that $|\beta|^e=|\pi_0|$. Then there exists an element $\pi$ of order one in $\mathfrak \pi$ s.t. one of the roots of the equation $X^e-\pi=0$ is contained in $K(\beta )$.
I don't see that if $E/K$ is a finite extension then $E/K$ is a totally ramified extension as the proof claims.