Norm Principles for Forms of Higher Degree Permitting Composition

Let F be a field of characteristic 0 or greater than d. Scharlau’s norm principle holds for separable field extensions K over F , for certain forms φ of degree d over F which permit composition. Introduction Let d ≥ 2 be an integer and let F be a field of characteristic 0 or > d. Let φ : V → F be a form of degree d on an F -vector space V of dimension n (i.e., after suitable identification, φ is a homogeneous polynomial of degree d in n indeterminates). Let K/F be a finite field extension of degree m. Scharlau’s norm principle (SNP) says that if a is a similarity factor of φK , then nK/F (a) is a similarity factor of φ. Knebusch’s norm principle (KNP) states that if a is represented by φF , then nK/F (a) is a product of m elements represented by φ, hence lies in the subgroup of F generated by DF (φ). Both norm principles were proved for nondegenerate quadratic forms over fields of characteristic not 2 (cf. [Sch, II.8.6] or [L, p. 205, p. 206]). For finite extensions of semi-local regular rings containing a field of characteristic 0, Knebusch’s norm principle (for quadratic forms) was proved in [Z] and for finite etale extensions of semi-local Noetherian domains with infinite residue fields of characteristic different from 2 in [O-P-Z]. We prove Scharlau’s norm principle for certain nondegenerate forms φ of degree d ≥ 3 which permit composition. Scharlau’s and Knebusch’s norm principle “coincide” for these forms, since they permit composition in the sense of Schafer [S] and thus satisfy DK(φ) = GK(φ) for all field extensions K/F . We explicitly compute the norms of similarity factors, if φ is the norm of a separable field extension or of a central simple algebra. 1. Preliminaries A form of degree d over F is a map φ : V → F on a finite-dimensional vector space V over F such that φ(av) = adφ(v) for all a ∈ F , v ∈ V and such that the 2000 Mathematics Subject Classification. Primary: 11E76; Secondary: 11E04, 12E05.