  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.
