Inmathematics, aglobal field is one of two types offields (the other one islocal fields) that are characterized usingvaluations. There are two kinds of globalfields:^[1]

• Algebraic number field: Afinite extension of${\displaystyle \mathbb{Q}}$
• Global function field: Thefunction field of anirreduciblealgebraic curve over afinite field, equivalently, a finite extension of${\displaystyle {\mathbb{F}}_q(T)}$, the field ofrational functions in one variable over the finite field with${\displaystyle q=p^n}$ elements.

An axiomatic characterization of these fields viavaluation theory was given byEmil Artin and George Whaples in the 1940s.^[2][3]

We say that field${\displaystyle K}$ isglobal field when there exists a set${\displaystyle \mathfrak{M}}$ of places (equivalence classes of absolute values on${\displaystyle K}$) such that:

• For every nonzero element${\displaystyle \alpha \in K}$,${\displaystyle |\alpha {|}_{\mathfrak{p}}=1}$ for all but finitely many${\displaystyle \mathfrak{p}\in \mathfrak{M}}$, and${\displaystyle \prod _{\mathfrak{p}\in \mathfrak{M}}|\alpha {|}_{\mathfrak{p}}=1}$
• At least one of places in${\displaystyle \mathfrak{M}}$ is either discrete with finite residue field or archimedean.

Only two kind of fields satisfy the axiomatic definition:

An algebraic number field

An algebraic number fieldF is a finite (and hencealgebraic)field extension of thefield ofrational numbersQ. ThusF is a field that containsQ and has finitedimension when considered as avector space overQ.

The function field of an irreducible algebraic curve over a finite field

A function field of analgebraic variety is the set of all rational functions on that variety. On an irreducible algebraic curve (i.e. a one-dimensional varietyV) over a finite field, we define a rational function on an open affine subsetU as the ratio of two polynomials in theaffine coordinate ring ofU, and a rational function on all ofV consists of such local data that agree on the intersections of open affine subsets. This technically defines the rational functions onV to be thefield 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 global fields. A field of either type has the property that all of itscompletions arelocally compact fields (seelocal fields). Every field of either type can be realized as thefield of fractions of aDedekind domain in which every non-zeroideal is of finite index. In each case, one has theproduct formula for non-zero elementsx:

${\displaystyle \prod _v|x{|}_v=1,}$

wherev varies over allvaluations of the field.

The analogy between the two kinds of fields has been a strong motivating force inalgebraic number theory. The idea of an analogy between number fields andRiemann surfaces goes back toRichard Dedekind andHeinrich M. Weber in the nineteenth century. The more strict analogy expressed by the 'global field' idea, in which a Riemann surface's aspect as algebraic curve is mapped to curves defined over a finite field, was built up during the 1930s, culminating in theRiemann hypothesis for curves over finite fields settled byAndré Weil in 1940. The terminology may be due to Weil, who wrote hisBasic Number Theory (1967) in part to work out the parallelism.

It is usually easier to work in the function field case and then try to develop parallel techniques on the number field side. The development ofArakelov theory and its exploitation byGerd Faltings in his proof of theMordell conjecture is a dramatic example. The analogy was also influential in the development ofIwasawa theory and theMain Conjecture. The proof of thefundamental lemma in theLanglands program also made use of techniques that reduced the number field case to the function field case.

TheHasse–Minkowski theorem is a fundamental result innumber theory that states that twoquadratic forms over a global field are equivalent if and only if they are equivalentlocally at all places, i.e. equivalent over everycompletion of the field.

Artin's reciprocity law implies a description of theabelianization of the absoluteGalois group of a global fieldK that is based on theHasse local–global principle. It can be described in terms of cohomology as follows:

LetL_v/K_v be aGalois extension oflocal fields with Galois groupG. Thelocal 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 thelocal Artin symbol, thelocal reciprocity map or thenorm residue symbol.^[4][5]

LetL/K be aGalois extension of global fields andC_L stand for theidèle class group ofL. The mapsθ_v for different placesv ofK can be assembled into a singleglobal symbol map by multiplying the local components of an idèle class. One of the statements of theArtin reciprocity law is that this results in a canonical isomorphism.^[6][7]

• Field (mathematics)
• Algebraic number theory
• Algebraic curves

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