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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.^[ref]

For an algebraic torus $T$ over $k$ and a chosen separable closure $k_s$, its geometric character lattice of an algebraic torus isIt is a free abelian group of rank $\dim T$, with its natural action of $\mathrm{Gal}(k_s/k)$. 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 $X^{\ast }(T)\cong {\mathbb{Z}}^n$. 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.^[ref]

A $k$-defined cocharacter of an algebraic torus $T$ is a homomorphism of algebraic groupsover $k$. A geometric cocharacter is defined after passing to a separable closure; the cocharacter lattice of an algebraic torus uses these geometric maps with their Galois action. The $k$-defined maps are the Galois-fixed elements.

For $T={\mathbb{G}}_m^2$, an integer pair $(a,b)$ gives $\lambda (t)=(t^a,t^b)$. Composing with the character $\chi (x,y)=x^m{y}^n$ yields $\chi (\lambda (t))=t^{ma+nb}$, defining the integer character-cocharacter pairing. This makes the two geometric lattices dual.

Cocharacters describe algebraic one-parameter subgroups and the integral directions underlying toric cones. For instance, $t\mapsto (t^2,t^3)$ has primitive exponent vector $(2,3)$; Bézout recovers $t$ from the coordinates by $t^3/t^2$. The distinction between geometric and ground-field-defined cocharacters matters for nonsplit tori.

The geometric cocharacter lattice of an algebraic torus $T$ over $k$ isusing a separable closure $k_s$. Composition with an algebraic torus character gives a homomorphism ${\mathbb{G}}_m\to {\mathbb{G}}_m$, necessarily $z\mapsto z^a$ for an integer $a$. This defines the perfect pairingFor a split algebraic torus, the cocharacter with exponent vector $(n_1,\dots ,n_r)$ is $z\mapsto (z^{n_1},\dots ,z^{n_r})$ and the pairing is ${\sum }_i{m}_i{n}_i$. The two geometric toric lattices carry compatible Galois group actions; $k$-defined cocharacters are the fixed elements. This is the duality used for the toric lattice $N$ and its dual $M$.^[ref]

For a finite Galois extension $L/K$, multiplicative Hilbert theorem 90 says $H^1(\mathrm{Gal}(L/K),L^{\ast })=0$. A multiplicative cocycle has the form $c_{\sigma }=\sigma (b)/b$. One proof forms a nonzero sum $A={\sum }_{\sigma }{c}_{\sigma }\sigma (a)$, using linear independence of field automorphisms. Its transformation law is $\tau A=c_{\tau }^{-1}A$, so $b=A^{-1}$ gives $c_{\tau }=\tau (b)/b$. Passing to a separable closure gives $H^1(G_K,K_s^{\times })=0$. For $m$ prime to the characteristic of a field, the multiplicative Kummer sequence then identifies $H^1(G_K,{\mu }_m)$ with $K^{\times }/(K^{\times }{)}^m$. The characteristic restriction makes the power map surjective on the separable closure.

Let $E/K$ be an elliptic curve with $\mathrm{char}K\ne 3$. Its geometric 3-torsion is a two-dimensional vector space over ${\mathbb{F}}_3$. Choosing a basis gives the continuous Galois representationHere $G_K$ is the absolute Galois group, acting on torsion coordinates in a separable closure. Its fixed vectors are exactly $E(K)[3]$. Thus a nonzero fixed vector means a rational point of order three; a stable one-dimensional vector subspace need only give a Galois-stable cyclic subgroup, not a rational generator.

The Weil pairing identifies ${\bigwedge }^{2}E[3]$ with ${\mu }_3$, so $\det {\rho }_{E,3}$ is the mod-three cyclotomic character. In particular, the image lies in ${\mathrm{SL}}_2({\mathbb{F}}_3)$ if $K$ contains all third roots of unity.^[ref]

In characteristic three, geometric 3-torsion is not ${\mathbb{F}}_3^2$. The group scheme $E[3]$ and its nonreduced structure require a different formulation.