Primes and The Field of Values of Characters
Nguyen N. Hung, Gabriel Navarro, Pham Huu Tiep
TL;DR
This work extends the field-of-values classification for irreducible characters to characters whose degrees are divisible by powers of a prime $p$, introducing and leveraging the conductor $c(\mathbb F)$ and the invariant $a=\nu_p(c(\mathbb F))$. It proves an existence theorem (Theorem \ref{thm:main1}) showing that for an abelian field $\mathbb F$ and any $b\ge \nu_p([\mathbb F_{p^a}:\mathbb F])$, there is a finite group $G$ with $\mathbb Q(\chi)=\mathbb F$ and $\nu_p(\chi(1))=b$, linking the $p$-part of character degrees to cyclotomic extensions; and, under a central conjecture, provides a precise criterion for when $\mathbb F$ can be realized with a given $p$-part. The paper furnishes extensive evidence across several natural families (e.g., $p$-solvable groups, symmetric/alternating groups, $\mathrm{GL}_n(q)$ and $\mathrm{GU}_n(q)$, $\mathrm{SL}_n(q)$, $\mathrm{SU}_n(q)$) and develops a general going-up/going-down framework under the inductive Feit condition, culminating in a broad divisibility result (subject to BKNT25) that ties $c(\chi)[{\mathbb Q}_{c(\chi)}:{\mathbb Q}(\chi)]$ to $|G|$ and yields a criterion for realizing abelian fields as character fields with prescribed $p$-parts. The work thus advances understanding of how arithmetic of fields of values interacts with the representation theory of finite groups beyond the $p'$-degree case and connects to foundational conjectures in modular representation theory and Galois actions.
Abstract
Let $p$ be a prime. For $p=2$, the fields of values of the complex irreducible characters of finite groups whose degrees are not divisible by $p$ have been classified; for odd primes $p$, a conjectural classification has been proposed. In this work, we extend this conjecture to characters whose degrees are divisible by arbitrary powers of $p$, and we provide some evidence supporting its validity.