Self-Invariant Maximal Subfields and Their Connexion with Some Conjectures in Division Rings

Abstract

Let D be a division algebra with center F . A maximal subfield of D is defined to be a fieldThis kind of subfields is important because they have strong connexion with most famous Albert's Conjecture (every division ring of prime index is cyclic). In fact, we pose a question that asserts whether every division ring whose all maximal subfields are self-invariant has to be commutative. The positive answer to this question, in finite dimensional case, implies the Albert's Conjecture (see §2). Although we show the Mal'cev-Neumann division ring demonstrates negative answer in the case of infinite dimensional division rings, but it is still most likely the question receives positive answer if we restrict ourselves to the finite dimensional division rings. We also have had the opportunity to use the Mal'cev-Neumann structure to answer Conjecture 1 below in negative (see §3). Finally, among other things, we rely on this kind of subfields to present a criteria for a division ring to have finite dimensional subdivision ring (see §4).