= Algebraic torus
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= Algebraic tori
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An <algebraic torus> over a <field> $k$ is an <algebraic group> $T$ whose <base change> to a <separable closure> $k_s$ is isomorphic to $\mathbb G_m^n$. Its <dimension> is $n$. An <algebraic torus> already isomorphic to $\mathbb G_m^n$ over $k$ is a <split algebraic torus>. <Toric geometry> described by an ordinary <toric fan> uses a <split algebraic torus>.
The real <algebraic group> $x^2+y^2=1$, with multiplication inherited from complex multiplication, is a one-dimensional nonsplit example. Over the <complex numbers>, $z=x+iy$ identifies its <coordinate ring> with $\mathbb C[z,z^{-1}]$. Complex conjugation acts on its geometric <character lattice of an algebraic torus> by $m\mapsto-m$, so it has no nontrivial real-defined <algebraic torus character>. <Algebraic tori> of <dimension> one therefore need not be the real multiplicative <group>.<Algebraic Groups, Lie Groups, and their Arithmetic Subgroups by James S. Milne>[chapterI, section14, Definition14.24 and character modules of tori]{ref}
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