Automorphism and similarity groups of forms determined by the characteristic polynomial

Abstract

The paper is concerned with forms of degree higher than two. M.Oryan and D.B.Shapiro [OS] proved that if A is a central simple algebra over a field K, ’tr‘ denotes its reduced trace and d ≥ 3, then every similarity of the trace form tr(Xd ) of degree d on A is standard. Now let A be a commutative and étale K— algebra of degree n ≥ 3. The coefficients of the generic characteristic polynomial χ A determine the norm form N A of degree n, the trace form of degree d, where d ≥ 3, and the intermediate forms M A,t of degree t, 2 ≤ t ≤ n - 1. We prove that all similarities of the intermediate forms are standard. The main result of the paper is the complete description of the similarity and the automorphism groups of the norm, trace and intermediate forms.