Fields with almost small absolute Galois group

TL;DR

This paper constructs specific fields with infinitely many degree-n extensions while maintaining finiteness in certain multiplicative groups across all finite extensions, revealing new structural properties of such fields.

Contribution

It introduces a method to construct fields with infinitely many degree-n extensions and finite multiplicative group quotients across all finite extensions.

Findings

Fields with infinitely many degree-n extensions are constructed.

Finiteness of E*/(E*)^n holds for all finite extensions E.

New structural properties of these special fields are established.

Abstract

We construct and study fields F with the property that F has infinitely many extensions of some fixed degree, but E*/(E*)^n is finite for every finite extension E of F and every n>0.