Height of Some Automorphisms of Local Fields

TL;DR

This paper characterizes certain automorphism subgroups of local fields, identifying those associated with specific types of p-adic extensions, thereby advancing understanding of local field automorphisms and their extension structures.

Contribution

It determines which automorphism subgroups correspond to _p-extensions and _p _p-extensions of characteristic zero local fields.

Findings

Identifies automorphism subgroups linked to _p-extensions.

Classifies automorphism subgroups related to _p _p-extensions.

Provides criteria for automorphism subgroup correspondence.

Abstract

In this note, we determine which automorphism subgroups of $\mathrm{Aut}_{\mathbb{F}_q}(\mathbb{F}_q((x)))$ are corresponding to $\mathbb{Z}_p$-extensions or $\mathbb{Z}_p\times\mathbb{Z}_p$-extensions of characteristic $0$ fields.