Maximal subfields in division algebras generated by images of polynomials

TL;DR

This work addresses when a maximal subfield of a finite-dimensional division ring $D$ over its center $F$ can be generated by a single element arising from a polynomial or a group word. The authors develop a framework using Laurent polynomial identities and algebraic-degree techniques to show that for any non-central multilinear polynomial $f$ or non-trivial word $w$, there exist evaluations yielding $F(f(a_1, olinebreak dots))$ or $F(w(a_1, olinebreak dots))$ as a maximal subfield, with degree $m= ext{dim}_F(F(a))^{1/2}$. Key constructions of matrices $P_m$ and $Q_m$ demonstrate that such values lie in the image of $f$ or $w$ and have algebraic degree $m$, enabling a dimension-theoretic conclusion. As an application, if all evaluations are algebraic over $F$ of bounded degree $d$, then $ ext{dim}_F D\le d^2$, extending Jacobson-type results and connecting maximal subfields to global structural bounds on $D$.

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 $D$.

Theorems & Definitions (22)

• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 3.1
• proof
• Lemma 3.2
• proof