The Galois group of a multidimensional local field of positive characteristic

O. V. Mel'nikov ; A. A. Sharomet

Sbornik. Mathematics, Tome 67 (1990) no. 2, pp. 595-610

Résumé

Let $K$ be an arbitrary field, Henselian relative to a discrete valuation $v$ of finite rank $n$ with residue field $k$. If $v=v_n\circ v_{n-1}\circ\dots\circ v_1$, where $v_i$ ($i=1,2,\dots,n$) is a discrete valuation of rank $1$, then, setting $K_n=K$, we denote by $K_{i-1}$ the residue field of the valuation $v_i$ of the field $K_i$, where $i=1,2,\dots,n$. A description of the absolute Galois group $\mathfrak G(K)$ of the field $K$, the inertia group $\mathfrak G^0(K)$ and the ramification group $\mathfrak G^1(K)$ of the valuation $v$ are obtained in terms of the absolute Galois group of the field of residues, its action on the roots of unity in the separable closure of the field $k$, and the cardinalities of the fields $K_0=k$ and $K_1,\dots,K_{n-1}$. Bibliography: 12 titles.