The term global field refers to either of the following:

• a number field, i.e., a finite extension of Q or
• the function field of an algebraic curve over a finite field, i.e., a finitely generated field of characteristic p>0 of transcendence degree 1.

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 quotient field 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: In mathematics, an algebraic number field (or simply number field) is a finite field extension of the rational numbers Q. That is, it is a field which contains Q and has finite dimension when considered as a vector space over Q. The study of algebraic number fields, and these days... In abstract algebra, a field extension L /K is called algebraic if every element of L is algebraic over K, i. ... In algebraic geometry, the function field of an irreducible algebraic variety is the field of fractions of the ring of regular functions. ... In algebraic geometry, an algebraic curve is an algebraic variety of dimension equal to 1. ... In abstract algebra, a finite field or Galois field (so named in honor of Evariste Galois) is a field that contains only finitely many elements. ... The word characteristic has several meanings: In mathematics, see characteristic (algebra) characteristic function characteristic subgroup Euler characteristic method of characteristics In genetics, see characteristic (genetics). ... In abstract algebra, the transcendence degree of a field extension L / K is a certain rather coarse measure of the size of the extension. ... In mathematics and related technical fields, a mathematical object is complete if nothing needs to be added to it. ... In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. ... In mathematics, a local field is a special type of field which has the additional property that it is a complete metric space with respect to a discrete valuation. ... In mathematics, every integral domain can be embedded in a field; the smallest field which can be used is the quotient field or the field of fractions of the integral domain. ... In abstract algebra, a Dedekind domain is a Noetherian integral domain which is integrally closed in its fraction field and which has Krull dimension 1. ... In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. ...

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v

The analogy between the two kinds of fields has been a strong motivating force in algebraic number theory. In particular, it is usually easier to work in the function field case and then try to develop parallel techniques on the number field side. The development of Arakelov theory and its exploitation by Gerd Faltings in his proof of the Mordell conjecture is a dramatic example. The function field analogue of the Riemann hypothesis is known to be true (by work of André Weil and others), and there is great interest in developing parallel techniques for number fields. In mathematics, an algebraic number field (or simply number field) is a finite field extension of the rational numbers Q. That is, it is a field which contains Q and has finite dimension when considered as a vector space over Q. The study of algebraic number fields, and these days... Gerd Faltings (born 28 July 1954) is a German mathematician known for his work in arithmetic algebraic geometry. ... In number theory, the Mordell conjecture stated a basic result regarding the rational number solutions to Diophantine equations. ... In mathematics, the Riemann hypothesis (aka Riemann zeta hypothesis), first formulated by Bernhard Riemann in 1859, is one of the most famous of all unsolved problems. ... André Weil (May 6, 1906 - August 6, 1998) was one of the great mathematicians of the 20th century, a founding member of the influential Bourbaki group. ...