In mathematics, the term global field refers to either of the following:
- a number field, i. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, an algebraic number field (or simply number field) F is a finite (and hence algebraic) Field extension of the e. , a finite extension of Q or
- the function field of an algebraic curve over a finite field, i. In Abstract algebra, a Field extension L / K is called algebraic if every element of L is algebraic over K, i In Algebraic geometry, an algebraic curve is an Algebraic variety of dimension one In Abstract algebra, a finite field or Galois field (so named in honor of Évariste Galois) is a field that contains only finitely many elements e. , a finitely generated field of characteristic p>0 of transcendence degree 1. In Mathematics, the characteristic of a ring R, often denoted char( R) is defined to be the smallest number of times one must add the ring's In Abstract algebra, the transcendence degree of a Field extension L / K is a certain rather coarse measure of the "size" of the extension
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). In Mathematical analysis, a Metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has In Topology and related branches of Mathematics, a Topological space is called locally compact if roughly speaking each small portion of the space looks In Mathematics, a local field is a special type of field that is a Locally compact Topological field with respect to a non-discrete topology 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 Mathematics, every Integral domain can be embedded in a field; the smallest field which can be used is the field of fractions or field of quotients In Abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an Integral domain in which every nonzero Proper In Ring theory, a branch of Abstract algebra, an ideal is a special Subset of a ring. In each case, one has the product formula for non-zero elements x:
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∏ | x | v = 1. v
The analogy between the two kinds of fields has been a strong motivating force in algebraic number theory. In Mathematics, algebraic number theory is a major branch of Number theory which studies the Algebraic structures related to Algebraic integers The idea of an analogy between number fields and Riemann surfaces goes back to Richard Dedekind and Heinrich M. Weber in the nineteenth century. In Mathematics, particularly in Complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional Julius Wilhelm Richard Dedekind ( October 6, 1831 &ndash February 12, 1916) was a German mathematician who did important Heinrich Martin Weber ( 5 March 1842 &ndash 17 May 1913) was a German Mathematician who specialized in Algebra The 19th century of the Common Era began on January 1, 1801 and ended on December 31, 1900, according to the Gregorian calendar The more strict analogy expressed by the 'global field' idea, in which a Riemann surface's aspect as algebraic curve is mapped to curves defined over a finite field, was built up during the 1930s, culminating in the Riemann hypothesis for local zeta-functions settled by André Weil in 1940. In Algebraic geometry, an algebraic curve is an Algebraic variety of dimension one André Weil should not be confused with two other mathematicians with similar names Hermann Weyl (1885-1955 who made substantial contributions The terminology may be due to Weil, who wrote his Basic Number Theory (1967) in part to work out the parallelism.
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. Arakelov theory (or Arakelov geometry) is an approach to Diophantine geometry, named for Suren Arakelov. Gerd Faltings (born July 28, 1954 in Gelsenkirchen -Buer is a German Mathematician known for his work in arithmetic Algebraic In Number theory, the Mordell conjecture stated a basic result regarding the rational number solutions to Diophantine equations It was eventually proved by Gerd
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