nLab global field

Definition

A global field is either

• a number field

• or a function field over a finite field;

hence is a finite-dimensional field extension of either

• the rational numbers $\mathbb{Q}$;

• the rational functions $\mathbb{F}_q(t)$ with coefficients in the finite field $\mathbb{F}_q$.

The function field analogy says how both these kinds of global fields are indeed analogous in that they behave as rings of rational functions on “arithmetic curves over the would-be field with one element $\mathbb{F}_1$” and $\mathbb{F}_q$, respectively. In particular for both types there is a concept of ring of adeles, group of ideles, etc.

The Langlands conjectures concern the arithmetic geometry of global fields.

The “local” version of a global field in the sense of formal geometry is a local field. The corresponding version of the Langlands program are the local Langlands correspondences.

Artin-Whaples characterization

Global fields can also be described axiomatically, as those fields which satisfy a suitable product formula. For example, if $k$ is a number field and $\mathbb{I}_k$ is its group of ideles, then there is an idele norm map

$$N: \mathbb{I}_k \to \mathbb{C}^\times$$

$$(x_v)_{places\; v} \mapsto \prod_v {|x_v|}_v$$

and $k^\times$ is retrieved as the kernel of this map, i.e., is the subgroup of elements $(x_v)$ for which the product formula $\prod_v {|x_v|}_v = 1$ holds.

For the following theorem, assume that $k$ is a field for which there is at least one valuation that is either archimedean, or is discrete and with finite residue class field for the corresponding local ring. By a place we understand an equivalence class of valuations.

Theorem

(Artin-Whaples) Suppose $k$ is a field that has a set of valuations ${|(-)|}_v$, one for every place of $k$, such that for every $x \in k^\times$ the product formula $\prod_v {|x|}_v = 1$ is satisfied. Then $k$ is a global field (and conversely, every global field has this property).

References

• Wikipedia, Global field

• Emil Artin and George Whaples, Axiomatic characterization of fields by the product formula for valuations, Bull. Amer. Math. Soc. 51 (1945), 469-492. (link to article)