Global field

Analogies between the two classes of fields

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 completions are locally compact fields (see local fields). Every field of either type can be realized as the field of fractions of a Dedekind domain in which every non-zero ideal is of finite index. In each case, one has the product formula for non-zero elements x:

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

where v varies over all valuations of the field.

The analogy between the two kinds of fields has been a strong motivating force in algebraic number theory. The idea of an analogy between number fields and Riemann surfaces goes back to Richard Dedekind and Heinrich 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 the Riemann hypothesis for curves over finite fields settled by André Weil in 1940. The terminology may be due to Weil, who wrote his Basic 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 of Arakelov theory and its exploitation by Gerd Faltings in his proof of the Mordell conjecture is a dramatic example. The analogy was also influential in the development of Iwasawa theory and the Main Conjecture. The proof of the fundamental lemma in the Langlands program also made use of techniques that reduced the number field case to the function field case.

Theorems

Hasse–Minkowski theorem

The Hasse–Minkowski theorem is a fundamental result in number theory that states that two quadratic forms over a global field are equivalent if and only if they are equivalent locally at all places, i.e. equivalent over every completion of the field.

Artin reciprocity law

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.

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.