value group of $E＝\Bbb{Q}_p(p^{1/e})$

Question

I want to find what is a value group of $E＝\Bbb{Q}_p(p^{1/e})$($e$ is positive integer, and this is totally ramified extension of degree $p$). I know value group of $K＝ \Bbb{Q}_p$ is {$p^a$｜$a∈\Bbb{Z}$}$∪${$0$}, I know extension of value is given by $|a|_E := \sqrt[n]{|N_{E/K}(a)|_K}$, but I'm having trouble calculating norm, could you tell me what is the value group of $E$ ?

Answer 1

$|p|=p^{-1}$ gives that $|p^{1/e}| = p^{-1/e}$. also $\Bbb{Q}_p(p^{1/e})=\bigoplus_{n=0}^{e-1} p^{n/e}\Bbb{Q}_p$ so that $|\sum_{n=0}^{e-1} a_n p^{n/e}|= \min_n |a_n| p^{-n/e}$ from which the value group is $p^{\Bbb{Z}/e}$.