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

TL;DR

This paper investigates self-invariant maximal subfields in division rings, exploring their connection to Albert's Conjecture, and examines conditions under which division rings are commutative or have finite-dimensional substructures.

Contribution

It introduces the concept of self-invariant maximal subfields, links them to Albert's Conjecture, and provides criteria for finite-dimensionality in division rings.

Findings

Negative answer in infinite-dimensional case via Mal'cev-Neumann division ring

Positive correlation between all maximal subfields being self-invariant and commutativity in finite dimensions

Criteria established for finite-dimensional subdivision rings

Abstract

Let D be a division algebra with center F. A maximal subfield of D is defined to be a field K such that CD(K) = K; that is, K is its own centralizer in D. A maximal subfield K is said to be self-invariant if it normalises by itself, i.e. ND*(K)= K: This kind of subfields is important because they have strong connexion with the 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 x2). 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 x3). 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 x4).