 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 found the following 2-adic counterexample for "d" = 4.Define

Then "G" has the property that it is 1 mod 4 if some "x" is odd, and 0 mod 16 otherwise.It follows easily from this that the homogeneous form

of degree "d" = 4 in 18 > "d"2 variables has no non-trivial zeros over the 2-adic integers.