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Ax–Kochen theorem

Index Ax–Kochen theorem

The Ax–Kochen theorem, named for James Ax and Simon B. Kochen, states that for each positive integer d there is a finite set Yd of prime numbers, such that if p is any prime not in Yd then every homogeneous polynomial of degree d over the p-adic numbers in at least d2+1 variables has a nontrivial zero. [1]

20 relations: Acta Arithmetica, Brauer's theorem on forms, Characteristic (algebra), Elsevier, Emil Artin, Finite field, Guy Terjanian, Hensel's lemma, James Ax, Jan Denef, Jean-Louis Colliot-Thélène, Laurent series, Mathematical logic, Model theory, P-adic number, Quasi-algebraically closed field, Serge Lang, Simon B. Kochen, Ultraproduct, Valuation (algebra).

Acta Arithmetica

Acta Arithmetica is a scientific journal of mathematics publishing papers on number theory.

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Brauer's theorem on forms

In mathematics, Brauer's theorem, named for Richard Brauer, is a result on the representability of 0 by forms over certain fields in sufficiently many variables.

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Characteristic (algebra)

In mathematics, the characteristic of a ring R, often denoted char(R), is defined to be the smallest number of times one must use the ring's multiplicative identity (1) in a sum to get the additive identity (0) if the sum does indeed eventually attain 0.

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Elsevier

Elsevier is an information and analytics company and one of the world's major providers of scientific, technical, and medical information.

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Emil Artin

Emil Artin (March 3, 1898 – December 20, 1962) was an Austrian mathematician of Armenian descent.

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Finite field

In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements.

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Guy Terjanian

Guy Terjanian is a French mathematician who has worked on algebraic number theory.

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Hensel's lemma

In mathematics, Hensel's lemma, also known as Hensel's lifting lemma, named after Kurt Hensel, is a result in modular arithmetic, stating that if a polynomial equation has a simple root modulo a prime number, then this root corresponds to a unique root of the same equation modulo any higher power of, which can be found by iteratively "lifting" the solution modulo successive powers of.

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James Ax

James Burton Ax (10 January 1937 – 11 June 2006) was a mathematician who proved several results in algebra and number theory by using model theory.

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Jan Denef

Jan Denef (born 4 September 1951) is a Belgian mathematician.

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Jean-Louis Colliot-Thélène

Jean-Louis Colliot-Thélène (born 2 December 1947), is a French mathematician.

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Laurent series

In mathematics, the Laurent series of a complex function f(z) is a representation of that function as a power series which includes terms of negative degree.

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Mathematical logic

Mathematical logic is a subfield of mathematics exploring the applications of formal logic to mathematics.

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Model theory

In mathematics, model theory is the study of classes of mathematical structures (e.g. groups, fields, graphs, universes of set theory) from the perspective of mathematical logic.

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P-adic number

In mathematics, the -adic number system for any prime number extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems.

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Quasi-algebraically closed field

In mathematics, a field F is called quasi-algebraically closed (or C1) if every non-constant homogeneous polynomial P over F has a non-trivial zero provided the number of its variables is more than its degree.

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Serge Lang

Serge Lang (May 19, 1927 – September 12, 2005) was a French-born American mathematician and activist.

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Simon B. Kochen

Simon Bernhard Kochen (born 14 August 1934, Antwerpen) is a Canadian mathematician, working in the fields of model theory, number theory and quantum mechanics.

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Ultraproduct

The ultraproduct is a mathematical construction that appears mainly in abstract algebra and in model theory, a branch of mathematical logic.

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Valuation (algebra)

In algebra (in particular in algebraic geometry or algebraic number theory), a valuation is a function on a field that provides a measure of size or multiplicity of elements of the field.

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Redirects here:

Ax-Kochen isomorphism theorem, Ax-Kochen theorem, Model theory of valued fields.

References

[1] https://en.wikipedia.org/wiki/Ax–Kochen_theorem

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