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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_{d} of prime numbers, such that if p is any prime not in Y_{d} then every homogeneous polynomial of degree d over the padic numbers in at least d^{2}+1 variables has a nontrivial zero.^{[1]}
The proof of the theorem makes extensive use of methods from mathematical logic, such as model theory. One first proves Serge Lang's theorem, stating that the analogous theorem is true for the field Fp((t)) of formal Laurent series over a finite field Fp with . In other words, every homogeneous polynomial of degree d with more than d^{2} variables has a nontrivial zero (so F_{p}((t)) is a C_{2} field). Then one shows that if two Henselian valued fields have equivalent valuation groups and residue fields, and the residue fields have characteristic 0, then they are equivalent (which means that a first order sentence is true for one if and only if it is true for the other). Next one applies this to two fields, one given by an ultraproduct over all primes of the fields F_{p}((t)) and the other given by an ultraproduct over all primes of the padic fields Q_{p}. Both residue fields are given by an ultraproduct over the fields F_{p}, so are isomorphic and have characteristic 0, and both value groups are the same, so the ultraproducts are equivalent. (Taking ultraproducts is used to force the residue field to have characteristic 0; the residue fields of F_{p}((t)) and Q_{p} both have nonzero characteristic p.) The 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_{p}((t)) if and only if it is true for the field of padic numbers. Applying this to the sentence stating that every nonconstant homogeneous polynomial of degree d in at least d^{2}+1 variables represents 0, and using Lang's theorem, one gets the AxKochen theorem. Alternative proof In 2008, Jan Denef found a purely geometric proof for a conjecture of JeanLouis ColliotThélène which generalizes the AxKochen theorem. He presented his proof at the "Variétés rationnelles" seminar [2] at École Normale Supérieure in Paris, but the proof has not been published yet. Exceptional primes Emil Artin conjectured this theorem without the finite exceptional set Yd, but Guy Terjanian[3] found the following 2adic counterexample for d = 4. Define G(x) = G(x_{1}, x_{2}, x_{3}) =Σ x_{i}^{4} − Σ_{i<j} x_{i}^{2}x_{j}^{2} − x_{1}x_{2}x_{3}(x_{1} + x_{2} + x_{3}). 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 G(x) + G(y) + G(z) + 4G(u) + 4G(v) + 4G(w) of degree d=4 in 18> d2 variables has no nontrivial zeros over the 2adic integers. Later Terjanian^{[4]} showed that for each prime p and multiple d>2 of p(p−1), there is a form over the padic numbers of degree d with more than d^{2} variables but no nontrivial zeros. In other words, for all d> 2, Y_{d} contains all primes p such that p(p−1) divides d. See also * Artin's conjecture
1. ^ James Ax and Simon Kochen, Diophantine problems over local fields I., American Journal of Mathematics, 87, pages 605630, (1965)
* Chang, C.C.; Keisler, H. Jerome (1989). Model Theory (third edition ed.). Elsevier. ISBN 0720406927. (Corollary 5.4.19) Retrieved from "http://en.wikipedia.org/"

