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Symmetric bilinear Forms and Galois Theory

Sugata Mandal

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 $σ_i \in G$. 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 $\frac{(n+1)}{2}$ constant rank $n$-subspaces, each having dimension of $n$. This holds particularly when $G$ is cyclic, represented as $G = \gal(L/K) = \langleσ\rangle$. 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^σ$ 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.

Symmetric bilinear Forms and Galois Theory

Abstract

Let be a field admitting a Galois extension of degree , denoting the Galois group as . Our focus lies on the space of symmetric -bilinear forms on . We establish a decomposition of into direct sum of -subspaces , where . Notably, these subspaces exhibit nice constant rank properties. The central contribution of this paper is a decomposition theorem for , revealing a direct sum of constant rank -subspaces, each having dimension of . This holds particularly when is cyclic, represented as . For cyclic extensions of even degree , we present slightly less precise but analogous results. In this scenario, we enhance and enrich these constant results and show that, the component often decomposes directly into a constant rank subspaces. Remarkably, this decomposition is universally valid when . Consequently, we derive a decomposition of into subspaces of constant rank under several situations. Moreover, leveraging these decompositions, we investigate the maximum dimension of an -subspace inside and for various field where and denote the vector spaces matrices and symmetric matrices over , respectively.
Paper Structure (9 sections, 18 theorems, 62 equations)

This paper contains 9 sections, 18 theorems, 62 equations.

Table of Contents

  1. Introduction
  2. Symmetric bilinear forms and Galois extensions
  3. Some Results on Cyclic Galois extension
  4. A short survey on $n$-subspace of $M_{n}(K)$ and $S_{n}(K)$
  5. For algebraic closed field
  6. For real number field
  7. For algebraic number field
  8. For finite field
  9. Conclusion

Key Result

Lemma 2.1

Let $\phi = \phi_{b,\sigma}$ be a symmetric bilinear form defined above. Thus we have for all $x$ and $y$ in $L$.

Theorems & Definitions (36)

  • Lemma 2.1
  • Corollary 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Remark 2.1
  • Definition 2.1
  • ...and 26 more