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| The '''Ax–Kochen theorem''', named for [[James Ax]] and [[Simon B. Kochen]], states that for each positive integer ''d'' there is a finite set ''Y<sub>d</sub>'' of prime numbers, such that if ''p'' is any prime not in ''Y<sub>d</sub>'' then every homogeneous polynomial of degree ''d'' over the [[p-adic number]]s in at least ''d''<sup>2</sup>+1 variables has a nontrivial zero.<ref>James Ax and Simon Kochen, ''Diophantine problems over local fields I.'', American Journal of Mathematics, '''87''', pages 605-630, (1965)</ref>
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| ==The proof of the theorem==
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| The proof of the theorem makes extensive use of methods from [[mathematical logic]], such as [[model theory]].
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| One first proves [[Serge Lang]]'s theorem, stating that the analogous theorem is true for the field '''F'''<sub>''p''</sub>((''t'')) of formal [[Laurent series]] over a [[finite field]] '''F'''<sub>''p''</sub> with <math>Y_d = \varnothing</math>. In other words, every homogeneous polynomial of degree ''d'' with more than ''d''<sup>2</sup> variables has a non-trivial zero (so '''F'''<sub>''p''</sub>((''t'')) is a [[quasi-algebraically closed field|C<sub>2</sub> field]]).
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| Then one shows that if two [[Hensel's lemma|Henselian]] [[Valuation (algebra)|valued]] fields have equivalent valuation groups and residue fields, and the residue fields have [[characteristic (algebra)|characteristic]] 0, then they are elementarily equivalent (which means that a first order sentence is true for one if and only if it is true for the other).
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| Next one applies this to two fields, one given by an [[ultraproduct]] over all primes of the fields '''F'''<sub>''p''</sub>((''t'')) and the other given by an ultraproduct over all primes of the ''p''-adic fields ''Q''<sub>''p''</sub>.
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| Both residue fields are given by an ultraproduct over the fields '''F'''<sub>''p''</sub>, so are isomorphic and have characteristic 0, and both value groups are the same, so the ultraproducts are elementarily equivalent. (Taking ultraproducts is used to force the residue field to have characteristic 0; the residue fields of '''F'''<sub>''p''</sub>((''t''))
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| and ''Q''<sub>''p''</sub> both have non-zero characteristic ''p''.)
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| The elementary equivalence of these ultraproducts implies that for any sentence in the language of valued fields, there is a finite set ''Y'' of exceptional primes, such that for any ''p'' not in this set the sentence is true for '''F'''<sub>''p''</sub>((''t'')) if and only if it is true for the field of ''p''-adic numbers. Applying this to the sentence stating that every non-constant homogeneous polynomial of degree ''d'' in at least ''d''<sup>2</sup>+1 variables represents 0, and using Lang's theorem, one gets the Ax–Kochen theorem.
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| ==Alternative proof==
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| In 2008, [[Jan Denef]] found a purely geometric proof for a conjecture of [[Jean-Louis Colliot-Thélène]] which generalizes the Ax–Kochen theorem. He presented his proof at the "Variétés rationnelles" seminar <ref>http://www.dma.ens.fr/~gille/sem/sem_variete_07-08.html</ref> at École Normale Supérieure in Paris, but the proof<ref>Jan Denef, [http://perswww.kuleuven.be/~u0009256/Denef-Mathematics/denef_papers/ColliotTheleneConjecture.pdf Proof of a conjecture of Colliot-Thélène]</ref> has not been published yet.
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| ==Exceptional primes==
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| [[Emil Artin]] conjectured this theorem with the finite exceptional set ''Y<sub>d</sub>'' being empty (that is, that all ''p''-adic fields are [[C2 field|C<sub>2</sub>]]), but [[Guy Terjanian]]<ref>{{cite journal | first=Guy | last=Terjanian | authorlink=Guy Terjanian | title=Un contre-example à une conjecture d'Artin | journal=C. R. Acad. Sci. Paris Sér. A-B | volume=262 | page=A612 | year=1966 | zbl=0133.29705 | language=French }}</ref> found the following 2-adic counterexample for ''d'' = 4. Define
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| :''G''('''x''') = ''G''(''x''<sub>1</sub>, ''x''<sub>2</sub>, ''x''<sub>3</sub>) =Σ ''x''<sub>i</sub><sup>4</sup> − Σ<sub>''i''<''j''</sub> ''x''<sub>i</sub><sup>2</sup>''x''<sub>j</sub><sup>2</sup> − ''x''<sub>1</sub>''x''<sub>2</sub>''x''<sub>3</sub>(''x''<sub>1</sub> + ''x''<sub>2</sub> + ''x''<sub>3</sub>).
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| Then ''G'' has the property that it is 1 mod 4 if some ''x'' is odd, and 0 mod 16 otherwise. It follows easily from this that the homogeneous form
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| :''G''('''x''') + ''G''('''y''') + ''G''('''z''') + 4''G''('''u''') + 4''G''('''v''') + 4''G''('''w''')
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| of degree ''d''=4 in 18> ''d''<sup>2</sup> variables has no non-trivial zeros over the 2-adic integers.
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| Later Terjanian<ref>Guy Terjanian, ''Formes ''p''-adiques anisotropes.'' (French) Journal für die Reine und Angewandte Mathematik, '''313''' (1980), pages 217-220</ref> showed that for each prime ''p'' and multiple ''d''>2 of ''p''(''p''−1), there is a form over the ''p''-adic numbers of degree ''d'' with more than ''d''<sup>2</sup> variables but no nontrivial zeros. In other words, for all ''d''> 2, ''Y<sub>d</sub>'' contains all primes ''p'' such that ''p''(''p''−1) divides ''d''.
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| {{harvtxt|Brown|1978}} gave an explicit but very large bound for the exceptional set of primes ''p''. If the degree ''d'' is 1, 2, or 3 the exceptional set is empty. {{harvtxt|Heath-Brown|2010}} showed that if ''d''=5 the exceptional set is bounded by 13, and {{harvtxt|Wooley|2008}} showed that for ''d''=7 the exceptional set is bounded by 883 and for ''d''=11 it is bounded by 8053.
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| ==See also==
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| *[[Artin's conjecture]]
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| *[[Brauer's theorem on forms]]
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| *[[quasi-algebraic closure]]
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| ==Notes==
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| <references/>
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| ==References==
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| *{{Citation | last1=Brown | first1=Scott Shorey | title=Bounds on transfer principles for algebraically closed and complete discretely valued fields | url=http://books.google.com/books?id=MesICi8orQkC | isbn=978-0-8218-2204-3 | mr=494980 | year=1978 | journal=Memoirs of the American Mathematical Society | issn=0065-9266 | volume=15 | issue=204}}
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| * {{ cite book | last=Chang | first=C.C. | author2-link=Howard Jerome Keisler|last2=Keisler|first2= H. Jerome | publisher=[[Elsevier]] | title=Model Theory | year=1989 | edition=third | isbn=0-7204-0692-7 }} (Corollary 5.4.19)
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| *{{Citation | last1=Heath-Brown | first1=D. R. | title=Zeros of p-adic forms | doi=10.1112/plms/pdp043 | mr=2595750 | year=2010 | journal=Proceedings of the London Mathematical Society. Third Series | issn=0024-6115 | volume=100 | issue=2 | pages=560–584}}
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| *{{Citation | last1=Wooley | first1=Trevor D. | title=Artin's conjecture for septic and unidecic forms | doi=10.4064/aa133-1-2 | mr=2413363 | year=2008 | journal=[[Acta Arithmetica]] | issn=0065-1036 | volume=133 | issue=1 | pages=25–35}}
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| {{DEFAULTSORT:Ax-Kochen theorem}}
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| [[Category:Number theory]]
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| [[Category:Model theory]]
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| [[Category:Theorems in number theory]]
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