= Character lattice of an algebraic torus
{title2=$M$}
For an <algebraic torus> $T$ over $k$ and a chosen <separable closure> $k_s$, its geometric <character lattice of an algebraic torus> is
$$
X^*(T)=\operatorname{Hom}_{k_s\text{-groups}}(T_{k_s},\mathbb G_{m,k_s}).
$$
It is a <free abelian group> of rank $\dim T$, with its natural action of $\operatorname{Gal}(k_s/k)$. <Characters of an algebraic torus> defined over $k$ form the fixed <subgroup> of this action; they need not constitute the full geometric <toric lattice>.
For a <split algebraic torus> $T=\mathbb G_m^n$, each <algebraic torus character> is a <Laurent monomial> $t_1^{m_1}\cdots t_n^{m_n}$, so $X^*(T)\cong\mathbb Z^n$. The real <norm-one algebraic torus> has geometric <algebraic torus character> <toric lattice> $\mathbb Z$ with complex conjugation acting by minus one, so its <subgroup> of real-defined <characters of an algebraic torus> is zero.<Algebraic Groups, Lie Groups, and their Arithmetic Subgroups by James S. Milne>[chapterI, section14, groups of multiplicative type and tori]{ref}
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