Maximal subfields in division algebras generated by images of polynomials

TL;DR

This paper proves that in a division algebra, maximal subfields can be generated by evaluating certain polynomials or words at elements of the algebra, revealing new ways to construct these subfields.

Contribution

It establishes that maximal subfields in division algebras can be generated by polynomial or word expressions evaluated at elements of the algebra, a novel insight.

Findings

Existence of elements generating maximal subfields via polynomial evaluations

Existence of elements generating maximal subfields via word evaluations

Maximal subfields can be constructed from polynomial and word expressions

Abstract

Let $D$ be a division ring with center $F$, $f(x_1,x_2,\dots, x_m)$ a non-central multilinear polynomial over $F$, and $w(x_1,x_2,\dots,x_m)$ a non-trivial word. In this paper, we investigate conditions under which there exists an element $a \in D$ such that the subfield $F(a)$ generated by $a$ is a maximal subfield of $D$. Specifically, we prove that there always exists an element $a$ in the set \[ \{f(a_1,\dots,a_m)\mid a_1,\dots, a_m\in D \} \cup \{w(a_1,\dots,a_m)\mid a_1,\dots, a_m\in D \backslash \{0\} \} \] such that $F(a)$ is a maximal subfield of $D$. This result shows that maximal subfields can be generated by evaluating polynomial or group word expressions at elements of $D$.