2007/09/05 by Rod Gow, Gow, Rod, Rachel Quinlan +1
Mathematics · #11E39 #12F10 #15A63 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Commutative Algebra (math.AC) #FOS: Mathematics #Finite Group Theory Research
paper · pdf · doi:10.48550/arxiv.0709.0741
openalex publication_date 2007/09/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.