Free fields in skew fields

TL;DR

This paper establishes conditions under which certain division rings contain free fields, with applications to division rings from torsion-free nilpotent groups and skew Laurent series.

Contribution

It provides new sufficient conditions for the existence of free fields within completed division rings, extending prior work by Chiba.

Findings

Division rings with discrete valuation can contain free fields under certain conditions.

Applications include division rings generated by torsion-free nilpotent groups.

Results extend the understanding of free fields in skew and Laurent series division rings.

Abstract

Building on the work of K. Chiba (J. Algebra 263 (2003), 75-87), we present sufficient conditions for the completion of a division ring with respect to the metric defined by a discrete valuation function to contain a free field, i.e. the universal field of fractions of a free associative algebra. Several applications to division rings generated by torsion-free nilpotent groups, skew Laurent series and related division rings are discussed.