Global field

In mathematics, a global field is one of two types of fields (the other one is local fields) that are characterized using valuations. There are two kinds of global fields:^[1]

• Algebraic number field: A finite extension of ${\displaystyle \mathbb {Q} }$
• Global function field: The
  irreducible algebraic curve over a finite field
  , equivalently, a finite extension of ${\displaystyle \mathbb {F} _{q}(T)}$, the field of rational functions in one variable over the finite field with ${\displaystyle q=p^{n}}$ elements.

An axiomatic characterization of these fields via valuation theory was given by Emil Artin and George Whaples in the 1940s.^[2][3]

A global field is one of the following:

An algebraic number field

An algebraic number field F is a finite (and hence

A function field of an algebraic variety is the set of all rational functions on that variety. On an irreducible algebraic curve (i.e. a one-dimensional variety V) over a finite field, we say that a rational function on an open affine subset U is defined as the ratio of two polynomials in the affine coordinate ring of U, and that a rational function on all of V consists of such local data that agree on the intersections of open affines. This technically defines the rational functions on V to be the field of fractions of the affine coordinate ring of any open affine subset, since all such subsets are dense.

There are a number of formal similarities between the two kinds of fields. A field of either type has the property that all of its

where v varies over all valuations of the field.

The analogy between the two kinds of fields has been a strong motivating force in

The

Artin's reciprocity law implies a description of the abelianization of the absolute Galois group of a global field K that is based on the Hasse local–global principle. It can be described in terms of cohomology as follows:

Let L_v/K_v be a Galois extension of local fields with Galois group G. The local reciprocity law describes a canonical isomorphism

${\displaystyle \theta _{v}:K_{v}^{\times }/N_{L_{v}/K_{v}}(L_{v}^{\times })\to G^{\text{ab}},}$

called the local Artin symbol, the local reciprocity map or the norm residue symbol.^[4][5]

Let L/K be a Galois extension of global fields and C_L stand for the idèle class group of L. The maps θ_v for different places v of K can be assembled into a single global symbol map by multiplying the local components of an idèle class. One of the statements of the Artin reciprocity law is that this results in a canonical isomorphism.^[6][7]