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 Q_p.

algebraic curve | class field | field | function field | Hasse principle | local field | number field | Riemann surface