Is there an infinite field whose multiplicative group is indecomposable?

TL;DR

This paper proves that for several major classes of infinite fields, their multiplicative groups are decomposable, supporting the conjecture that no infinite field has an indecomposable multiplicative group.

Contribution

It confirms the conjecture for finitely generated, local, global, and function fields, expanding the understanding of multiplicative group structures in infinite fields.

Findings

Indecomposable multiplicative groups do not occur in these classes of infinite fields.

The conjecture holds for finitely generated fields, local fields, and global fields.

The paper extends previous results on finite fields to broader classes of infinite fields.

Abstract

In an earlier paper, we determined the finite fields with indecomposable multiplicative groups and conjectured that there is no infinite field whose multiplicative group is indecomposable. In this paper, we prove this conjecture for several popular classes of fields, including finitely generated fields, discrete valued fields, fields of Hahn series, local fields, global fields, and function fields.