Global body

Global solids are the central subjects of study in the mathematical branch of algebraic number theory . The best known global body is that of the rational numbers . On the other hand, local bodies are created by completing global bodies.

definition

One calls a global body

• on the one hand algebraic number fields , d. H. finite extensions of the field of rational numbers . They have zero characteristics. ${\displaystyle \mathbb {Q} }$
• and on the other hand algebraic function fields of positive characteristic of degree of transcendence 1, i.e. H. finite extensions of for a prime and an indefinite . ${\displaystyle \mathbb {F} _{p}(T)}$ ${\displaystyle p}$ ${\displaystyle T}$

The completions of global bodies at each point with respect to their respective metrics are local bodies. The fact that both number fields and function fields are global fields expresses an analogy between number and function fields known since the 19th century ( Richard Dedekind et al.). For the more difficult number field case, this often makes it possible to work with methods that were developed in the function field case and have a natural geometric interpretation there.

Axiomatic characterization according to Artin and Whaples

Let be a field with a set of primes such that the following axioms are fulfilled. ${\displaystyle K}$ ${\displaystyle {\mathfrak {V}}}$

• For everyone is for almost everyone and it applies (product formula). ${\displaystyle a\in K^{\times }}$ ${\displaystyle |a|_{v}=1}$ ${\displaystyle v\in {\mathfrak {V}}}$ ${\displaystyle \prod _{v\in {\mathfrak {V}}}|a|_{v}=1}$
• There is one such that there is a local body . ${\displaystyle v\in {\mathfrak {V}}}$ ${\displaystyle K_{v}}$

Then is a global field and consists of all primes of . ${\displaystyle K}$ ${\displaystyle {\mathfrak {V}}}$ ${\displaystyle K}$

Web links

• William Stein, Global Fields