Symmetric bilinear Forms and Galois Theory

2024/02/07 by Mandal, Sugata
#Commutative Algebra (math.AC) #FOS: Mathematics #Number Theory (math.NT) #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.2402.04604

Abstract

Let K be a field admitting a Galois extension L of degree n, denoting the Galois group as G = \gal(L/K). Our focus lies on the space \sym_K(L) of symmetric K-bilinear forms on L. We establish a decomposition of \sym_K(L) into direct sum of K-subspaces A^σ_i, where σ_i ∈ G. Notably, these subspaces A^σ_i exhibit nice constant rank properties. The central contribution of this paper is a decomposition theorem for \sym_K(L), revealing a direct sum of ((n+1))/(2) constant rank n-subspaces, each having dimension of n. This holds particularly when G is cyclic, represented as G = \gal(L/K) = ⟨σ⟩. For cyclic extensions of even degree n = 2m, we present slightly less precise but analogous results. In this scenario, we enhance and enrich these constant results and show that, the component A^σ often decomposes directly into a constant rank subspaces. Remarkably, this decomposition is universally valid when -1 ∉ L². Consequently, we derive a decomposition of \sym_K(L) into subspaces of constant rank under several situations. Moreover, leveraging these decompositions, we investigate the maximum dimension of an n-subspace inside M(n,K) and S(n,K) for various field K where M(n,K) and S(n,K) denote the vector spaces (n × n) matrices and symmetric matrices over K, respectively.

Related

• Galois extensions and subspaces of bilinear forms with special rank properties
• Constant rank subspaces of alternating bilinear forms from Galois Theory
• Dimension bounds for constant rank subspaces of symmetric bilinear forms\n over a finite field
• Rank-related dimension bounds for subspaces of symmetric bilinear forms
• Connections between rank and dimension for subspaces of bilinear forms