Maximal subfields in division algebras generated by images of polynomials

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 $$\{f(a_1,\dots ,a_m)\mid a_1,\dots ,a_m\in D\}\cup \{w(a_1,\dots ,a_m)\mid a_1,\dots ,a_m\in D\mathrm{\backslash }\{0\}\}$$ 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 $D$.