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

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∈D satisfies a polynomial equation xn+αn−1xn−1+⋯+α0=0 with α0,…,αn−1∈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 d2-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.