Ax–Kochen theorem

Remarks

The proof is model-theoretic. Let $\varphi$ be a first-order sentence in the language of rings. Then $\varphi$ holds in $\Q_p$ for almost all primes $p$ if and only if it holds in the Laurent series field $\F_p((t))$ for almost all $p$. Here "almost all" means all but finitely many. This transfer comes from ultraproducts over a nonprincipal ultrafilter on the primes. The ultraproducts of the fields $\Q_p$ and of the fields $\F_p((t))$ are elementarily equivalent valued fields. Both are Henselian with residue field of characteristic $0$, and they have the same residue field and the same value group. For a fixed degree $d$, the property "every form of degree $d$ in $d^2 + 1$ variables has a nontrivial zero" is one first-order sentence. A theorem of Lang shows that it holds in every $\F_p((t))$. The finite exceptional set cannot be dropped in general: Terjanian found a form of degree $4$ in $18$ variables over $\Q_2$ with no nontrivial zero, while $18 > 4^2$.

Going deeper

For a student

The theorem is a celebrated application of model theory to number theory. The key principle is now called the Ax–Kochen–Ershov principle. It says that the first-order theory of a Henselian valued field of residue characteristic $0$ is determined by the theories of its residue field and value group. The original proof gives no explicit bound for the exceptional primes. For $d = 2$ and $d = 3$ the conclusion is known to hold for all primes.