Small skew fields

TL;DR

This paper proves that small skew fields in non-zero characteristic are necessarily commutative, extending known results about finite and certain infinite fields.

Contribution

It establishes that small skew fields in non-zero characteristic are commutative, confirming a conjecture posed by Wagner.

Findings

Small skew fields in non-zero characteristic are commutative.

Supports the conjecture that small fields are algebraically closed and commutative.

Extends classical results from finite and superstable fields to small skew fields.

Abstract

Wedderburn showed in 1905 that finite fields are commutative. As for infinite fields, we know that superstable (Cherlin, Shelah) and supersimple (Pillay, Scanlon, Wagner) ones are commutative. In their proof, Cherlin and Shelah use the fact that a superstable field is algebraically closed. Wagner showed that a small field is algebraically closed, and asked whether a small field should be commutative. We shall answer this question positively in non-zero characteristic.