Cubic Homogeneous Polynomials Over -Adic Number Fields

Abstract

It is known [11 that a homogeneous polynomial equation in n variables of a given fixed degree r over a p-adic number field has a non-trivial solution in that field provided n exceeds some sufficiently large number which depends on r; say n ? c(r). Moreover, many examples exist showing that 0(r) > r2. It has been shown by Hasse [4] that for the quadratic case n ? 5 is sufficient. We here settle the cubic case and prove the following: THEOREM. If K is a complete field under a discrete non-archimedian valuation and has a finite residue class field, then every cubic homogeneous polynomial equation in n variables with coefficients in K, has a non-trivial solution in K, provided n ? 10. Recently, Demyanov [3] has proved this result under the additional hypothesis that the characteristic of the residue class field is not three, an assumption we have been able to avoid. Our proof was arrived at independently at the same time and is quite different from that of Demyanov, although both proofs make use of a crucial result of Chevalley [2].