Assume K is a finite extension of $\mathbb{Q}$$_p$. Let W = K (d$^{\frac{1}{p^n}}$), d $\in$ K$^{\times}$. And $\phi_{W/K}$ is the Hasse-Herbrand function of W/K.
$\forall$ 1 $\leq$ j $\leq$ n, c$_j$ = je + $\frac{e}{p-1}$, in which e is the ramification index of K/$\mathbb{Q}$$_p$.
We also have U$_{K,1}^{(s)}$ = U$_{K}^{(s)}$ / (K$^{\times p}$ $\cap$ U$_{K}^{(s)}$), and suppose d $\in$ U$_{K,1}^{(t)}$\U$_{K,1}^{(t+1)}$ in which
1 $\leq$ t < c$_1$ and p does not divide t.
Question: how to compute $\phi_{W/K}$