On the Similarity Group of Forms of Higher Degree

Abstract

Let ƒ = ƒ(X) = ƒ(X1,..., Xn,) k[X1,...,Xn] be a non-singular form of degree d ≥ 3 over the field k (where char(k)= 0 or char(k) > d). The similarity group of ƒ, Sk(ƒ) is the subgroup of GLn(k) consisting of all matrices A such that ƒ(AX) = λAƒ(X) for some λA k×. Let Autk(ƒ) denote the automorphism group of ƒ. Identifying k× with the group of scalar matrices in Sk(ƒ), we show using Kummer′s theory of fields that Sk(ƒ)/Autk(ƒ)k× is a finite abelian group of exponent d. We apply this result to give a complete characterization of the similarity group of rational binaries of odd degree.