Inertia group

In number theory, more specifically in local class field theory, the ramification groups are a filtration of the Galois group of a local field extension, which gives detailed information on the ramification phenomena of the extension.

Ramification groups are a refinement of the Galois group ${\displaystyle G}$ of a finite ${\displaystyle L/K}$ Galois extension of local fields. We shall write ${\displaystyle w,{\mathcal {O}}_{L},{\mathfrak {p}}}$ for the valuation, the ring of integers and its maximal ideal for ${\displaystyle L}$. As a consequence of Hensel's lemma, one can write ${\displaystyle {\mathcal {O}}_{L}={\mathcal {O}}_{K}[\alpha ]}$ for some ${\displaystyle \alpha \in L}$ where ${\displaystyle O_{K}}$ is the ring of integers of ${\displaystyle K}$. (This is stronger than the primitive element theorem.) Then, for each integer ${\displaystyle i\geq -1}$, we define ${\displaystyle G_{i}}$ to be the set of all ${\displaystyle s\in G}$ that satisfies the following equivalent conditions.

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