 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 F"p"(("t")) 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 "d"2+1 variables represents 0, and using Lang's theorem, one gets the Ax–Kochen theorem.Jan Denef found a purely geometric proof for a conjecture of Jean-Louis Colliot-Thélène which generalizes the Ax–Kochen theorem.