Feit numbers and $p'$-degree characters

TL;DR

This paper investigates the relationship between the minimal cyclotomic field containing a character's values and the structure of solvable groups, establishing divisibility properties related to character degrees and group normalizers.

Contribution

It proves that for solvable groups with odd degree characters, the minimal cyclotomic field degree divides the order of a specific quotient of the normalizer, linking character values to group structure.

Findings

f_ ext{chi} divides |N_G(P)/P'| for certain characters

f_ ext{chi} equals the order of an element in N_G(P)/P'

establishes a connection between cyclotomic fields and group normalizers

Abstract

Suppose that $\chi$ is an irreducible complex character of $G$ and let $f_\chi$ be the smallest integer $n$ such that the cyclotomic field $\mathbb Q_n$ contains the values of $\chi$. Let $p$ be a prime, and assume that $\chi \in \textrm{Irr}(G)$ has degree not divisible by $p$. If $G$ is solvable and $\chi(1)$ is odd, then there exists $g \in {\bf N}_G(P) /P'$ with $o(g)=f_\chi$. In particular $f_\chi$ divides $|{\bf N}_G(P) /P'|$.