Symmetric bilinear Forms and Galois Theory

Let $K$ be a field admitting a Galois extension $L$ of degree $n$, denoting the Galois group as $G=\text{gal}(L\mathrm{/}K)$. Our focus lies on the space ${\text{sym}}_K(L)$ of symmetric $K$-bilinear forms on $L$. We establish a decomposition of ${\text{sym}}_K(L)$ 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 ${\text{sym}}_K(L)$, 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 $G=\text{gal}(L\mathrm{/}K)=⟨\sigma ⟩$. 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 $-1\mathrm{\notin }{L}^2$. Consequently, we derive a decomposition of ${\text{sym}}_K(L)$ 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.