Galois groups of p-extensions of higher local fields

Abstract

Suppose $\mathcal K$ is $N$-dimensional local field of characteristic $p$, $\mathcal G =\mathop{Gal}(\mathcal K_{sep}/\mathcal K)$, $\mathcal G_{<p}$ is the maximal quotient of $\mathcal G$ of period $p$ and nilpotent class $<p$ and $\mathcal K_{<p}\subset \mathcal K_{sep}$ is such that $\mathop{Gal}(\mathcal K_{<p}/\mathcal K)=\mathcal G_{<p}$. We use nilpotent Artin-Schreier theory to identify $\mathcal G_{<p}$ with the group $G(\mathcal L)$ obtained from a profinite Lie $\mathbb F_p$-algebra $\mathcal L$ via the Campbell-Hausdorff composition law. The canonical $\mathcal P$-topology on $\mathcal K$ is used to define a dense Lie subalgebra $\mathcal L^{\mathcal P}$ in $\mathcal L$. The algebra $\mathcal L^{\mathcal P}$ can be provided with a system of $\mathcal P$-topological generators and its $\mathcal P$-open subalgebras correspond to all $N$-dimensional extensions of $\mathcal K$ in $\mathcal K_{<p}$. These results are applied to higher local fields $K$ of characteristic 0 containing primitive $p$-th root of unity. If $\Gamma =\mathop{Gal}(K_{alg}/K)$ we introduce similarly the quotient $\Gamma_{<p}=G(L)$, a dense $\mathbb F_p$-Lie algebra $L^{\mathcal P}\subset L$, and describe the structure of $L^{\mathcal P}$ in terms of generators and relations. The general result is illustrated by explicit presentation of $\Gamma_{<p}$ modulo third commutators.