Maximal subfields in division algebras generated by images of polynomials

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 $D$.