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Ramification group

If $L/K$ is a finite <Galois extension> of local fields, $\mathcal O_L=\mathcal O_K[\alpha]$, and $f$ is the <minimal polynomial> of $\alpha$, then
$$
v_L(f'(\alpha))
=\sum_{1\ne\sigma\in\operatorname{Gal}(L/K)}v_L(\sigma(\alpha)-\alpha)
=\sum_{s\geq0}(|G_s|-1).
$$

Ramification-group sum for a monogenic integer ring

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 136 / 2 / b / Solution

Ramification-group sum for a monogenic integer ring

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Ramification-group sum for a monogenic integer ring

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