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 
, 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).
