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  "body": "\n## Definition\n\nLet $k$ be an algebraically closed field, and let $G$ be a smooth connected [[linear algebraic group|GA81VVQHEP]] over $k$.\nAn element $g \\in \\mathrm{GL}_n(k)$ is *unipotent* if $g - I$ is a [[nilpotent matrix|07BSPZPSMF]].\nA linear algebraic group is *unipotent* if all its points over $k$ are unipotent in some, equivalently every, faithful representation.\nThe *unipotent radical* $R_u(G)$ is the largest smooth connected unipotent [[normal subgroup|0CB0DX5BVV]] of $G$.\nThe group $G$ is *reductive* if $R_u(G)$ is trivial.\n\n## Remarks\n\nOver an arbitrary field $k$, a smooth connected linear algebraic group $G$ is called reductive if $G \\times_k \\bar{k}$ is reductive.\n\nExample: $\\mathrm{GL}_n$ is reductive.\nLet $U$ be a connected normal unipotent subgroup.\nBy Kolchin's theorem, the vectors fixed by $U$ form a nonzero subspace of $k^n$.\nSince $U$ is normal, $\\mathrm{GL}_n$ preserves this subspace, so it is all of $k^n$ and $U$ is trivial.\n\nNon-example: the group $B$ of invertible upper triangular $2 \\times 2$ matrices is not reductive.\nThe subgroup of matrices $\\begin{pmatrix} 1 \u0026 b \\\\ 0 \u0026 1 \\end{pmatrix}$ is normal in $B$, connected and unipotent.\n\nIn characteristic $0$, a smooth connected linear algebraic group is reductive if and only if all its finite-dimensional representations are semisimple.\n",
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