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I've asked this question in several more elementary forums, and haven't get any answer. So I presume this is not so an elementary question.

Let $L/K$ be a Galois extension, and $w$ be a valuation of a $L$, lying above a valuation $v$ of $K$. Notice that I do not suppose that $w$ is discrete. Given a real number $\alpha >0$ in the image of $w$, each of the following can easily been shown to be a subgroup of the inertia group of $w$ in $L$ :

  • $\{ \sigma\in Gal(L/K) : w(\sigma x - x) \geq \alpha \}$,

  • $\{ \sigma\in Gal(L/K) : w(\sigma x - x) > \alpha \}$,

  • $\{ \sigma\in Gal(L/K) : w(\sigma x - x) \geq w(x) + \alpha \}$,

  • $\{ \sigma\in Gal(L/K) : w(\sigma x - x) > w(x) + \alpha \}$.

What is the terminology for these subgroups ? (I guess some variant of "ramification group of order $\alpha$") ? Can you indicate me sources where these groups are dealt in ? Thx.

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  • $\begingroup$ KConrad has weakened my initial question regarding general valuations to valuations whose group of values is the set of real numbers. I believe his reediting of the question follows from the lack of answers in the general case. Nevertheless, I would be happy to get an answer even in the general case. $\endgroup$
    – MikeTeX
    Commented Dec 21, 2014 at 21:17

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