A note on Artin's diophantine conjecture

TL;DR: Artin this paper showed that every quadratic form in at least 5 variables over the field Q p of p-adic numbers has a nontrivial zero in Q p. This fact has led Artin to make the conjecture (C): "Every form over q p of degree d in n > d 2 variables has a non-trivial 0.

Abstract: A well known theorem of Hasse [1] says that every quadratic form in at least 5 variables over the field Q p of p-adic numbers has a nontrivial zero. This fact has led Artin to make the conjecture (C): \"Every form over Q p of degree d in n > d 2 variables has a non-trivial zero.\" However, a counterexample has been provided by Terjanian [2] in the case d=4. The case d=2 is distinguished by the fact that every quadratic form may be \"diagonalized\", i.e., assumed to be of the type Σ a i X 2 i . One is therefore led to the weaker conjecture (C): \"Every form f= Σ a i X d i over Q p in n > d 2 variables has a nontrivial zero in Q p,\" which still generalizes Hasse's theorem.