Split embedding problems over the open arithmetic disc

TL;DR

This paper proves that all finite split embedding problems over Q and t-unramified problems over the quotient field of O{t} can be solved within the ring of arithmetic power series converging on the open unit disc, extending classical Galois theory results.

Contribution

It extends Harbater's classical result by solving all finite split embedding problems over Q and t-unramified problems over the quotient field of O{t} for arbitrary number fields.

Findings

Every finite group occurs as a Galois group over the field of arithmetic power series.

All finite split embedding problems over Q are solvable over this field.

The results apply to the quotient field of O{t} for any number field K.

Abstract

Let Z{t} be the ring of arithmetic power series that converge on the complex open unit disc. A classical result of Harbater asserts that every finite group occurs as a Galois group over the quotient field of Z{t}. We strengthen this by showing that every finite split embedding problem over Q acquires a solution over this field. More generally, we solve all t-unramified finite split embedding problems over the quotient field of O{t}, where O is the ring of integers of an arbitrary number field K.