I am studying higher ramification groups of local field extension.
In Wikipedia it is mentioned that higher ramification groups gives information about ramification of extension.
Suppose $L/K$ be a Galois extension of local field with decreasing filtration $$G_{-1}:=G \supset G_0 \supset G_1 \supset \cdots $$ It is given that
$G_{0}$ is trivial iff $L/K$ is unramified
$G_1$ is trivial iff $L/K$ is tamely ramified.
I think $G_0=G$ iff $L/K$ is totally ramified.
So it seems the first $3$ filtrations of $G$ is enough to say whether a local field extension is totally ramified or unramified or tamely ramified or wildly ramified.
Ami I right ?
But I am lacking motivation how exactly the ramification groups help to study ramifications because it seems harder to compute $G_0, G_1, \cdots$ practically.
So my question would be-
Can you write (or list) some examples of local field extensions (eg., $p$-cyclotomic extension) where one can talk about ramifications by computing the ramification groups by hand ?
I indeed want to understand example of local field extension where one can compute the first few ramification groups ($G_{-1},G_0, G_1, \cdots)$ and thereby making conclusion whether the extension is unramified/totally ramified.
Thanks