On the Similarity Group of Forms of Higher Degree

Abstract

Let ƒ = ƒ( X) = ƒ( X 1,..., X n ,) ∈ k[ X 1,..., X n ] 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, S k (ƒ) is the subgroup of GL n ( k) consisting of all matrices A such that ƒ( AX) = λ A ƒ( X) for some λ A ∈ k ×. Let Aut k (ƒ) denote the automorphism group of ƒ. Identifying k × with the group of scalar matrices in S k (ƒ), we show using Kummer′s theory of fields that S k (ƒ)/Aut k (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.