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  "body": "\n## Definition\n\nLet $k$ be a field with algebraic closure $\\bar{k}$.\nLet $\\mathbb{G}_m = \\operatorname{Spec} k[t, t^{-1}]$ be the multiplicative group.\nAn *algebraic torus* over $k$ is an [[algebraic group|GH81SYAPPN]] $T$ over $k$ such that $T \\times_k \\bar{k} \\cong \\mathbb{G}_{m, \\bar{k}}^n$ as algebraic groups for some $n \\ge 0$.\nThe torus is *split* if $T \\cong \\mathbb{G}_m^n$ already over $k$.\n\n## Remarks\n\nAn algebraic torus is a [[linear algebraic group|GA81VVQHEP]], and $n$ is its dimension.\nOne can replace $\\bar{k}$ by a separable closure of $k$ in the definition.\n\nExample: over $k = \\R$, let $T$ be the circle $x^2 + y^2 = 1$ with $(x, y)(x', y') = (x x' - y y', x y' + x' y)$.\nOver $\\C$, the map $(x, y) \\mapsto x + iy$ is an isomorphism $T_{\\C} \\cong \\mathbb{G}_{m, \\C}$, so $T$ is a torus of dimension $1$.\nIt is not split, because $T(\\R)$ is compact while $\\mathbb{G}_m(\\R) = \\R^{*}$ is not.\n\nNon-example: the additive group $\\mathbb{G}_a = \\operatorname{Spec} k[x]$ is not a torus.\nThe ring $\\bar{k}[x]$ has only constant units, while $\\bar{k}[t, t^{-1}]$ has the unit $t$, so $\\mathbb{G}_{a, \\bar{k}} \\not\\cong \\mathbb{G}_{m, \\bar{k}}$.\n",
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