# nLab global field

### Context

#### Algebra

higher algebra

universal algebra

# Contents

## Definition

A global field is either

hence is a finite-dimensional field extension of either

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).

• 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)