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Concept# Global field

Summary

In mathematics, a global field is one of two type of fields (the other one is local field) which are characterized using valuations. There are two kinds of global fields:
Algebraic number field: A finite extension of
Global function field: The function field of an algebraic curve over a finite field, equivalently, a finite extension of , the field of rational functions in one variable over the finite field with elements.
An axiomatic characterization of these fields via valuation theory was given by Emil Artin and George Whaples in the 1940s.
Algebraic number field and Function field of an algebraic variety
A global field is one of the following:
An algebraic number field
An algebraic number field F is a finite (and hence algebraic) field extension of the field of rational numbers Q. Thus F is a field that contains Q and has finite dimension when considered as a vector space over Q.
The function field of an algebraic curve over a finite field
A function field of a variety is the set of all rational functions on that variety. On an algebraic curve (i.e. a one-dimensional variety V) over a finite field, we say that a rational function on an open affine subset U is defined as the ratio of two polynomials in the affine coordinate ring of U, and that a rational function on all of V consists of such local data which agree on the intersections of open affines. This technically defines the rational functions on V to be the field of fractions of the affine coordinate ring of any open affine subset, since all such subsets are dense.
There are a number of formal similarities between the two kinds of fields. A field of either type has the property that all of its completions are locally compact fields (see local fields). Every field of either type can be realized as the field of fractions of a Dedekind domain in which every non-zero ideal is of finite index. In each case, one has the product formula for non-zero elements x:
The analogy between the two kinds of fields has been a strong motivating force in algebraic number theory.

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