Top Artin slope for an extension of local fields (reviewed)

Let $L/K$ be an extension of $p$-adic fields with Artin slope content $[s_1, \ldots, s_k]_t^u$. The top Artin slope of $L/K$ is the largest $s$ such that the ramification group $G^{(s)}$ is nontrivial. Hence

• if $k>0$, then the top slope is $s_k$, which is always greater than $1$
• otherwise, if $t>1$, then the top slope is $1$
• otherwise the top slope is $0$

This includes the convention that the top slope of the trivial extension $K/K$ is $0$.