Global Field

A global field is either a number field or a finite extension field of , the field of rational functions over a finite field. Function fields of complex algebraic curves are analogous objects, but are not global fields in this arithmetic sense. A number field contains a canonical subring, its ring of integers. In the function field case, a choice of gives the polynomial ring and its integral closure in the global field, but this subring is not canonical. Choosing a nonzero prime ideal in either such subring determines a valuation at which the 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 .

The word "global" distinguishes the field from its local field completions. In the analogous setting of a complex algebraic curve, its function field consists of rational functions that may have poles. These are not necessarily functions defined at every point. Under favorable conditions, the local information can be patched together to yield global information (e.g., the Hasse principle).