# Relation between local and global inertia/ramification degrees

Let $$K/\mathbb{Q}$$ be a number field and suppose a prime $$p\in\mathbb{Z}$$ factors in $$\mathcal{O}_K$$ as $$\prod_{i=1}^r \mathfrak{p}_i^{e_i}$$. From algebraic number theory, we have the identity $$[K:\mathbb{Q}]=\sum_{i=1}^r e_if_i.$$ If $$v_{ \mathfrak{p}_i}$$ is the $$\mathfrak{p}_i$$-adic valuation on $$K$$ and $$K_{\mathfrak{p}_i}$$ the corresponding extension of $$\mathbb{Q}_p$$, is it true that $$[K_{ \mathfrak{p}_i}:\mathbb{Q}_p]=e_if_i?$$ I know a statement like this exists for local fields - i.e., if $$L/K$$ is an extension of local fields then $$[L:K]=ef$$ - but I'm wondering how these 'local' and 'global' statements interact... Any references would be welcome.

Yes your "natural intuition" is right here. $$e,f$$ can be defined using statements about the the decomposition group (and the inertia group), and the natural map $$D_{\mathfrak{p}_i/p} \rightarrow \textrm{Gal}(K_{\mathfrak{p}_i}/\mathbb{Q}_p)$$ is an isomorphism. (In general you can pass to a Galois closure and use the tower laws for $$e$$ and $$f$$.)