Free fields in Malcev-Neumann series rings

TL;DR

The paper demonstrates that Malcev-Neumann series rings often contain free fields of countable rank, using valuation-based criteria, with implications for skew Laurent series rings and embeddability of uncountable free fields.

Contribution

It introduces a criterion for embedding free fields into skew fields complete with respect to a valuation, and applies this to Malcev-Neumann series rings and related structures.

Findings

Malcev-Neumann series rings often contain free fields of countable rank.

A valuation-based criterion for embedding free fields is established.

Applications to skew Laurent series rings and uncountable free fields are discussed.

Abstract

It is shown that the skew field of Malcev-Neumann series of an ordered group frequently contains a free field of countable rank, i.e. the universal field of fractions of a free associative algebra of countable rank. This is an application of a criterion on embeddability of free fields on skew fields which are complete with respect to a valuation function, following K. Chiba. Other applications to skew Laurent series rings are discussed. Finally, embeddability questions on free fields of uncountable rank in Malcev-Neumann series rings are also considered.