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 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 p-adic numbers in at least d²+1 variables has a nontrivial zero.

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 F_p((t)) of formal Laurent series over a finite field F_p with ${\displaystyle Y_{d}=\varnothing }$. In other words, every homogeneous polynomial of degree d with more than d² variables has a non-trivial zero (so F_p((t)) is a C₂ 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 elementarily 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 p-adic 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 elementarily 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 non-zero characteristic p.)

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