Resumen de Shirshov�s theorem and division rings that are left algebraic over a subfield

Jason P. Bell, Vesselin Drensky, Yaghoub Sharifi

• LetDbe a division ring.Wesay thatDis left algebraic over a (not necessarily central) subfield K of D if every x �¸ D satisfies a polynomial equation xn + �¿n.1xn.1 + �E �E �E + �¿0 = 0 with �¿0, . . . , �¿n.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.Weconclude by giving a list of problems for algebras that are left algebraic over a commutative subring.