×

On the length of a quadratic form. (English) Zbl 1174.11375

Tandon, Rajat (ed.), Algebra and number theory. Proceedings of the silver jubilee conference, Hyderabad, India, December 11–16, 2003. New Delhi: Hindustan Book Agency (ISBN 81-85931-57-7/hbk). 147-157 (2005).
The goal of the paper is to take the next step towards relating two numerical invariants of a field \(k\), \(\text{char}(k)\neq 2\). The first one is the \(u\)-invariant: \(u(k)\) is the maximal dimension of an anisotropic quadratic form over \(k\), or \(\infty\) if there is no such number. The second one is a sequence \(\lambda_ n(k)\) of non-negative integers which describes the behaviour of mod 2 Galois cohomology of the field \(k\). A cohomology group \(H^ n(k,\mathbb{Z}/2)=H^ n(k,\mu_ 2)\) has a distinguished set of generators (\(n\)-symbols, i.e., cup products of elements of \(H^ 1(k,\mathbb Z/2)\)). \(\lambda_ n(k)\) is the minimal number of these generators such that an element of \(H^ n(k,\mathbb Z/2)\) is a sum of \(\leq\lambda_ n(k)\) of them, or \(\infty\) if there is no such number. The cohomology groups are related to quadratic forms by \(H^ n(k,\mathbb Z/2)\cong I^ n(k)/I^ {n+1}(k)\).
The authors recall known facts: the stronger condition that there is a number \(N\) such that \(I^ N(k)=0\) and \(\lambda_ i(k)\) finite for \(1<i<N\) implies \(u(k)\) finite; and \(u(k)\) finite implies existence of \(N\) such that \(I^ N(k)=0\).
B. Kahn [in: Algebra and number theory. Proceedings of the silver jubilee conference, Hyderabad, India, 2003. New Delhi: Hindustan Book Agency, 21–33 (2005; Zbl 1099.11020)] raised the question if finiteness of \(u(k)\) implies finiteness of all \(\lambda_ n(k)\). The affirmative answer is known for \(n=1\) (obvious) and \(n=2\).
The main result of the paper (Theorem 3.4) is that finiteness of \(u(k)\) implies finiteness of \(\lambda_ 3(k)\). Moreover, a general result is proven: if for every field extension \(K/k\) for every \(n\geq 3\) any element of \(H^ n(K,\mathbb Z/2)\) has a generic splitting variety, then the question of Kahn has an affirmative answer.
For the entire collection see [Zbl 1066.11001].

MSC:

11E81 Algebraic theory of quadratic forms; Witt groups and rings
12G05 Galois cohomology

Citations:

Zbl 1099.11020
PDFBibTeX XMLCite