The Ax–Kochen theorem, named for James Ax and Simon B. Kochen, states that for each positive integer d there is a finite set Yd of prime numbers, such that if p is any prime not in Yd then every homogeneous polynomial of degree d over the p-adic numbers in at least d2 + 1 variables has a nontrivial zero.
In other words, every homogeneous polynomial of degree d with more than d2 variables has a non-trivial zero (so Fp((t)) is a C2 field).
Jan Denef found a purely geometric proof for a conjecture of Jean-Louis Colliot-Thélène which generalizes the Ax–Kochen theorem.
[2][3] Emil Artin conjectured this theorem with the finite exceptional set Yd being empty (that is, that all p-adic fields are C2), but Guy Terjanian[4] found the following 2-adic counterexample for d = 4.
Later Terjanian[5] showed that for each prime p and multiple d > 2 of p(p − 1), there is a form over the p-adic numbers of degree d with more than d2 variables but no nontrivial zeros.