Algebraic torus

Remarks

An algebraic torus is a linear algebraic group, and $n$ is its dimension. One can replace $\bar{k}$ by a separable closure of $k$ in the definition.

Example: over $k = \R$, let $T$ be the circle $x^2 + y^2 = 1$ with $(x, y)(x', y') = (x x' - y y', x y' + x' y)$. Over $\C$, the map $(x, y) \mapsto x + iy$ is an isomorphism $T_{\C} \cong \mathbb{G}_{m, \C}$, so $T$ is a torus of dimension $1$. It is not split, because $T(\R)$ is compact while $\mathbb{G}_m(\R) = \R^{*}$ is not.

Non-example: the additive group $\mathbb{G}_a = \operatorname{Spec} k[x]$ is not a torus. The ring $\bar{k}[x]$ has only constant units, while $\bar{k}[t, t^{-1}]$ has the unit $t$, so $\mathbb{G}_{a, \bar{k}} \not\cong \mathbb{G}_{m, \bar{k}}$.