A global field is either a number field, a function field on an algebraic curve, or an extension of transcendence degree one over a finite field. From a modern point of view, a global field may refer to a function field on a complex algebraic curve as well as one over a finite field. A global field contains a canonical subring, either the algebraic integers or the polynomials. By choosing a prime ideal in its subring, a global field can be topologically completed to give a local field. For example, the rational numbers are a global field. By choosing a prime number p, the rationals can be completed in the p-adic norm to form the p-adic numbers .

A global field is called global because of the special case of a complex algebraic curve, for which the field consists of global functions (i.e., functions that are defined everywhere). These functions differ from functions defined near a point, whose completion is called a local field. Under favorable conditions, the local information can be patched together to yield global information (e.g., the Hasse principle).

 

Algebraic Curve, Class Field, Field, Function Field, Hasse Principle, Local Field, Number Field, Riemann Surface