Galois extensions and subspaces of alternating bilinear forms with special rank properties. (English) Zbl 1211.15030
Summary: Let \(K\) be a field admitting a cyclic Galois extension of degree \(n\). The main result of this paper is a decomposition theorem for the space of alternating bilinear forms defined on a vector space of odd dimension \(n\) over \(K\). We show that this space of forms is the direct sum of \((n-1)/2\) subspaces, each of dimension \(n\), and the non-zero elements in each subspace have constant rank defined in terms of the orders of the Galois automorphisms. Furthermore, if ordered correctly, for each integer \(k\) lying between 1 and \((n-1)/2\), the rank of any non-zero element in the sum of the first \(k\) subspaces is at most \(n-2k+1\). Slightly less sharp similar results hold for cyclic extensions of even degree.
MSC:
15A63 | Quadratic and bilinear forms, inner products |
12F10 | Separable extensions, Galois theory |
11E39 | Bilinear and Hermitian forms |
Keywords:
field; Galois extension; cyclic extension; Galois group; alternating bilinear form; constant rank; polynomialReferences:
[1] | Delsarte, P., Bilinear forms over a finite field, J. Combin. Theory Ser. A, 25, 226-241 (1978) · Zbl 0397.94012 |
[2] | Delsarte, P.; Goethals, J. M., Alternating bilinear forms over \(GF(q)\), J. Combin. Theory Ser. A, 19, 26-50 (1975) · Zbl 0343.05015 |
[3] | Gow, R.; Quinlan, R., On the vanishing of subspaces of alternating bilinear forms, Linear and Multilinear Algebra, 54, 415-428 (2006) · Zbl 1110.15028 |
[4] | Graham, W., Nonemptiness of skew-symmetric degeneracy loci, Asian J. Math., 9, 261-271 (2005) · Zbl 1094.14034 |
[5] | Ilic, B.; Landsberg, J. M.; loci, On symmetric degeneracy, spaces of matrices of constant rank and dual varieties, Math. Ann., 314, 159-174 (1999) · Zbl 0949.14028 |
[6] | Lang, S., Algebra (1993), Addison-Wesley: Addison-Wesley Reading, Mass · Zbl 0848.13001 |
[7] | Lidl, R.; Niederreiter, H., Finite Fields (1983), Addison-Wesley: Addison-Wesley Reading, Mass · Zbl 0554.12010 |
[8] | Meshulam, R., On two extremal matrix problems, Linear Algebra Appl., 114/115, 261-271 (1989) · Zbl 0665.15011 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.