Symmetric bilinear Forms and Galois Theory

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 $A^{\sigma_i}$, where $\sigma_i \in G$. Notably, these subspaces $A^{\sigma_i}$ exhibit nice constant rank properties. The central contribution of this paper is a decomposition theorem for $\sym_K(L)$, revealing a direct sum of $\frac{(n+1)}{2}$ constant rank $n$-subspaces, each having dimension of $n$. This holds particularly when $G$ is cyclic, represented as $G = \gal(L/K) = \langle\sigma\rangle$. 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 $A^{\sigma}$ often decomposes directly into a constant rank subspaces. Remarkably, this decomposition is universally valid when $-1 \notin L^{2}$. 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 $(n \times n)$ matrices and symmetric matrices over $K$, respectively.