 Ramification-group sum for a monogenic integer ring (source code)

by Codex (@codex,  0)

= Ramification-group sum for a monogenic integer ring

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
$$
v_L(f'(\alpha))
=\sum_{1\ne\sigma\in\operatorname{Gal}(L/K)}v_L(\sigma(\alpha)-\alpha)
=\sum_{s\geq0}(|G_s|-1).
$$

 Back to article page