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 fieldFp) of formal Laurent series over a finite fieldFp with. In other words, every homogeneous polynomial of degree d with more than d2 variables has a non-trivial zero ) is a C2 field). Then one shows that if two Henselianvalued fields have equivalent valuation groups and residue fields, and the residue fields have characteristic 0, then they are elementarily equivalent. Next one applies this to two fields, one given by an ultraproduct over all primes of the fields Fp) and the other given by an ultraproduct over all primes of the p-adic fields Qp. Both residue fields are given by an ultraproduct over the fields Fp, so are isomorphic and have characteristic 0, and both value groups are the same, so the ultraproducts are elementarily equivalent. ) and Qp both have non-zero characteristic p.) 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 Fp) 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 d2+1 variables represents 0, and using Lang's theorem, one gets the Ax–Kochen theorem.
Alternative proof
found a purely geometric proof for a conjecture of Jean-Louis Colliot-Thélène which generalizes the Ax–Kochen theorem.
conjectured this theorem with the finite exceptional set Yd being empty, but Guy Terjanian found the following 2-adic counterexample for d = 4. Define 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 of degree d = 4 in 18 > d2 variables has no non-trivial zeros over the 2-adic integers. Later Terjanian showed that for each prime p and multiple d > 2 of p, there is a form over the p-adic numbers of degree d with more than d2 variables but no nontrivial zeros. In other words, for all d > 2, Yd contains all primes p such that p divides d. 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. showed that if d = 5 the exceptional set is bounded by 13, and showed that for d = 7 the exceptional set is bounded by 883 and for d = 11 it is bounded by 8053.
