Let $K$ be a number field and let $\nu$ be a discrete valuation on $K$ (this might be, for example, the valuation attached to a prime ideal $\mathfrak{P}$ of $K$ ).

Let $K_{\nu}$ be the completion of $K$ at $\nu$ , and let $\mathcal{O}_{\nu}$ be the ring of integers of $K_{\nu}$ , i.e. $$\mathcal{O}_{\nu}=\{ k\in K_{\nu} \mid \nu(k)\geq 0 \}$$ The maximal ideal of $\mathcal{O}_{\nu}$ will be denoted by $$\mathcal{M}=\{ k\in K_{\nu} \mid \nu(k)>0 \}$$ and we denote by $k_{\nu}$ the residue field of $K_{\nu}$ , which is $$k_{\nu}=\mathcal{O}_{\nu}/\mathcal{M}$$ We will consider three different global Galois groups, namely $$G_{\overline{K}/K}=\operatorname{Gal}(\overline{K}/K)$$ $$G_{\overline{K_{\nu}}/K_{\nu}}=\operatorname{Gal}(\overline{K_{\nu}}/K_{\nu})$$ $$G_{\overline{k_{\nu}}/k_{\nu}}=\operatorname{Gal}(\overline{k_{\nu}}/k_{\nu})$$ where $\overline{K},\overline{K_{\nu}},\overline{k_{\nu}}$ are separable algebraic closures of the corresponding field. We also define notation for the inertia group of $G_{\overline{K_{\nu}}/K_{\nu}}$ $$I_{\nu} \subseteq G_{\overline{K_{\nu}}/K_{\nu}}$$

Remark: By Galois theory we know that, $K_{\nu}^{\operatorname{nr}}$ , the fixed field of $I_{\nu}$ , the inertia subgroup, is the maximal unramified extension of $K_{\nu}$ , so $$I_{\nu}\cong \operatorname{Gal}(\overline{K_{\nu}}/K_{\nu}^{\operatorname{nr}})$$