Shirshov's theorem and division rings that are left algebraic over a subfield

TL;DR

This paper proves that division rings left algebraic over a subfield of bounded degree are finite-dimensional over their center, extending Kaplansky's result, using a new combinatorial approach based on Shirshov's theorem.

Contribution

It generalizes Kaplansky's theorem by establishing a bound on the dimension over the center for division rings left algebraic over subfields, with a novel combinatorial proof.

Findings

Division rings left algebraic over a subfield of degree d are at most d^2-dimensional over their center.

A new combinatorial version of Shirshov's theorem is developed for words over finite alphabets.

The paper discusses open problems for algebras left algebraic over a commutative subring.

Abstract

Let D be a division ring. We say that D is left algebraic over a (not necessarily central) subfield K of D if every x in D satisfies a polynomial equation x^n + a_{n-1}x^{n-1}+...+a_0=0 with a_0,...,a_{n-1} in K. We show that if D is a division ring that is left algebraic over a subfield K of bounded degree d then D is at most d^2-dimensional over its center. This generalizes a result of Kaplansky. For the proof we give a new version of the combinatorial theorem of Shirshov that sufficiently long words over a finite alphabet contain either a q-decomposable subword or a high power of a non-trivial subword. We show that if the word does not contain high powers then the factors in the q-decomposition may be chosen to be of almost the same length. We conclude by giving a list of problems for algebras that are left algebraic over a commutative subring.