Daniel B. Shapiro

Witt Rings and Brauer Groups Under Multiquadratic Extensions .2.

Let F be a field of characteristic not 2, and suppose M is a multiquadratic extension of F. That is, M/F is a finite abelian extension of exponent 2, so that M = F(@) for some finite subgroup G E F/F’, where

Spaces of similarities. I. The Hurwitz problem

= F (0). In studying Brauer groups and products of quaternion algebras, the second author was led to consider the homology groups N,(M/F) of -a certain complex gM,F associated to the extension M/F. This complex appears in [ 11, (3.1); 301 and in (1.1) below. The purpose of the present work is to investigate the first homology group N,(M/F) and to exhibit its close connections with quadratic form theory. The higher homology groups are examined in [ 11,301. The field F is said to have property Pi(n) if N,(M/F) = 1 for every multiquadratic extension M/F with [M:F] Q 2”. Properties P,(l) and P,(2) always hold, but there are examples [29] of fields F for which P,(3) fails. These examples are generalized in Section 5.

On trace forms of higher degree

Abstract Contents . Introduction. l. Terminology and Sim( V ). 2. Upper bound functions and a tensor construction. 3. Clifford , algebras and similarity representations. 4. ( s, t )-families. 5. Commuting spaces of similarities. 6. Hermitian forms.

Multiplicative subgroups of index three in a field

Let ti be a central simple algebra over a field k, and let tr denote its reduced trace. Then qd( X) = tr X” is the trace form of degree d on &. A similarity of W, Vd) is a linear map f : d +d for which there exists A E k” such that

Centers of higher degree forms

Theorem. If G be a subgroup of index 3 in the multiplicative group F* of a field F,then G+G = F,except in the cases IF= 4, 7, 13, or 16. The elementary methods used here provide a new proof of the classical case when F is finite. If F is a finite field and IFI 54 4 or 7, then every element c E F can be expressed as a sum of two cubes: c = x3+y3 for some x, y E F. Furthermore such x, y exist with xy 54 0 in F provided IFI 54 4, 7, 13, 16. Versions of these results have appeared in various forms in the literature. For example, see [3 p. 95 and p. 104, 7, 8, and 9]. This theorem also follows from the known values of the cyclotomic numbers when e = 3, as given for example in [10, p. 35]. We present here a generalization to arbitrary fields. If F is a finite field where the multiplicative group F* has order divisible by 3, then the nonzero cubes F*3 form the unique subgroup of index 3 in F*. Theorem. Let G be a subgroup of index 3 in the multiplicative group F* of a field F. Then G+ G = F, except in the cases IFI = 4, 7, 13, or 16. The Theorem is proved in an elementary fashion, not using the classical results mentioned above. It is valid for fields of any cardinality and any characteristic.

Spaces of Similarities. II. Pfister Factors

Let be a symmetric d-linear form on a vector space V of dimension n over a field k. Its center, Cent(), is the analog of the space of symmetric matrices for a bilinear form. If d> 2, the center is a commutative subalgebra of End(V ). It seems difficult to determine which subalgebras can be realized as Cent() for some some d-linear form . As a first step we conjecture that the center has dimension at most n. The conjecture is proved for n 5.

On multiples of round and Pfister forms

Contents. Introduction. 7. The questions. 8. Using the involution. 9. Inside the Clifford algebra.