Ax-Kochen Theorem

The Ax-Kochen Theorem, named for James Ax and Simon B. Kochen, is a result proved using the model-theoretic technique of ultraproducts, comparing the p-adic numbers to formal power series fields. It states that for every positive integer d there is a finite set of exceptional prime numbers such that, for any prime p outside that set, every homogeneous polynomial of degree d in at least d squared plus one variables over the p-adic numbers has a nontrivial zero. The theorem settled a conjecture of Emil Artin apart from the finitely many exceptional primes it itself allows for.