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  "body": "\n## Definition\n\nLet $G$ be a finite [[group|G5S1M9]], let $p$ be a prime, and let $k$ be an [[algebraically closed field|0PF6TM8TWT]] of characteristic $p$.\nAn element $g \\in G$ is $p$-regular if its [[order|0WE2NHZJ7B]] is not divisible by $p$; let $G_{p'}$ be the set of $p$-regular elements.\nLet $\\mu_{p'}(k)$ and $\\mu_{p'}(\\C)$ be the groups of [[roots of unity|0YV2CKP2AV]] of order prime to $p$ in $k$ and in the [[complex numbers|09FTAET5Z4]].\nFix a group isomorphism $\\phi \\colon \\mu_{p'}(k) \\to \\mu_{p'}(\\C)$.\nLet $\\rho \\colon G \\to GL(V)$ be a [[representation|WYGRXCG2RN]] of $G$ on a vector space $V$ over $k$ of finite dimension $d$.\nFor $g \\in G_{p'}$, the map $\\rho(g)$ has [[eigenvalues|0BQAA2BK3Q]] $\\lambda_1, \\ldots, \\lambda_d \\in \\mu_{p'}(k)$, listed with multiplicity.\nThe *Brauer character* of $\\rho$ is the function $\\beta_\\rho \\colon G_{p'} \\to \\C$ given by\n$$\\beta_\\rho(g) = \\phi(\\lambda_1) + \\cdots + \\phi(\\lambda_d).$$\n\n## Remarks\n\nThe eigenvalues lie in $\\mu_{p'}(k)$ because $\\rho(g)^m = \\id$ with $m$ the order of $g$, prime to $p$.\nSuch an isomorphism $\\phi$ exists; it is usually fixed through a $p$-modular system, and other choices change $\\beta_\\rho$ by a Galois automorphism.\nExample: for the trivial representation of degree $d$, all eigenvalues are $1$, so $\\beta(g) = d$ for every $g \\in G_{p'}$; for $d = p$ this is $p$, while the trace in $k$ is $0$.\nNon-example: let $G = \\Z/p$ and let $V = k^2$, with the generator acting by $\\begin{pmatrix} 1 \u0026 1 \\\\ 0 \u0026 1 \\end{pmatrix}$.\nThe generator is not $p$-regular, so the Brauer character has no value there.\nIt is defined only at the identity, with value $2$.\nBrauer characters are constant on conjugacy classes, additive on short exact sequences, and $\\beta_\\rho(1) = \\dim V$.\nThe irreducible Brauer characters are linearly independent, and their number is the number of conjugacy classes of $p$-regular elements.\n",
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