Brauer character

《Cohomologie Locale Des Faisceaux Coherents (Sga 2): Seminaire De Geometrie Algebrique Du Bois Marie 1962 (Documents Mathematiques)》 · 印刷第 192 页 Section 10.1「The Definition of Brauer Characters」的模系与单位根约定、第 193 页 Lemma 10.1.1 后的提升记号及 Example 10.1.2 前的定义段（省略中间内容）

Whenever we work with Brauer characters for a finite group $G$ in characteristic $p$ we will implicitly assume that we have a $p$-modular system $(F,R,k)$ where both $F$ and $k$ are splitting fields for $G$ and all of its subgroups. This has the implication that $F$ and $k$ both contain a primitive $a$th root of unity, where $a$ is the l.c.m. of the orders of the $p$-regular elements of $G$ (the elements of order prime to $p$). […] We will write the bijection between $a$th roots of unity in $F$ and in $k$ as $\hat\lambda\mapsto\lambda$, so that if $\lambda$ is an $a$th root of unity in $k$ then $\hat\lambda$ is the root of unity in $R$ which maps onto it. […] Let $g\in G$ be a $p$-regular element, and let $\rho:G\to\operatorname{GL}(U)$ be a representation over $k$. Then $\rho(g)$ is diagonalizable, since $k\langle g\rangle$ is semisimple and all eigenvalues of $\rho(g)$ lie in $k$, being $a$th roots of unity. If the eigenvalues of $\rho(g)$ are $\lambda_1,\ldots,\lambda_n$ we put $\varphi_U(g)=\hat\lambda_1+\cdots+\hat\lambda_n$, and this is the Brauer character of $U$. It is a function that is only defined on the $p$-regular elements of $G$, and takes values in a field of characteristic zero, which we may always take to be $\mathbb C$.