Algebraic torus

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Definition

A torus (sometimes called an algebraic torus) over a field is an algebraic group that can be described as a direct product of finitely many multiplicative groups of finite extensions of that field. Note that we could use the multiplicative group of the field itself in one or more of the direct factors.

The term split torus is used for a torus that is a direct product of copies of the multiplicative group of the field. When the field is an algebraically closed field, any torus over it is a split torus.

This use of the word torus should not be confused with complex torus, which is a quotient of $\mathbb{C}^n$ by a lattice in it.

Examples

Finite field

For a finite field of size $q$, there is a unique field extension of degree $d$ for every positive integer $d$. We can thus classify the $n$-dimensional tori over the field by the set of unordered integer partitions of $n$. For each unordered integer partition $n = d_1 + d_2 + \dots + d_r$, the corresponding torus is:

$\mathbb{F}_{q^{d_1}}^\ast \times \mathbb{F}_{q^{d_2}}^\ast \times \dots \times \mathbb{F}_{q^{d_r}}^\ast$

The order of such a torus is:

$\prod_{i=1}^r (q^{d_i} - 1)$

Although all these groups have order approximately $q^n$, the exact orders are different for each torus.

Dimension Partition of dimension Corresponding torus Order
1 1 $\mathbb{F}_q^\ast$ $q - 1$
2 1 + 1 $\mathbb{F}_q^\ast \times \mathbb{F}_q^\ast$ -- split torus $(q - 1)^2$
2 2 $\mathbb{F}_{q^2}^\ast$ $q^2 - 1$
3 1 + 1 + 1 $\mathbb{F}_q^\ast \times \mathbb{F}_q^\ast\times \mathbb{F}_q^\ast$ -- split torus $(q - 1)^3$
3 2 + 1 $\mathbb{F}_{q^2}^\ast\times \mathbb{F}_q^\ast$ $(q^2 - 1)(q - 1)$
3 3 $\mathbb{F}_{q^3}^\ast$ $q^3 - 1$

Field of real numbers

The only possible field extensions of finite degree over the field of real numbers are the field of real numbers and the field of complex numbers. Thus, the tori are all direct products of copies of the multiplicative groups of these two fields. The following are the possibilities for tori of small dimension over the field of real numbers:

Dimension Partition of dimension as a sum of 1s and 2s Corresponding torus
1 1 $\R^\ast$, multiplicative group of the field of real numbers
2 1 + 1 $\R^\ast \times \R^\ast$. This is the split torus.
2 2 $\mathbb{C}^\ast$
3 1 + 1 + 1 $\R^\ast \times \R^\ast \times \R^\ast$. This is the split torus.
3 2 + 1 $\mathbb{C}^\ast \times \mathbb{R}^\ast$

Algebraically closed field

For an algebraically closed field, there is a unique torus in each dimension -- the split torus defined as the direct product of as many copies of the multiplicative group of the field as the dimension.