来自伽罗瓦理论的交错双线性形式常秩子空间
Constant rank subspaces of alternating bilinear forms from Galois Theory
AI总结：
本文针对次数为2m的循环扩张L/K上的交错双线性形式空间，改进常秩性质结果，证明分量A^σ常可分解为常秩子空间，还得到相关非退化子空间的存在性结论。
摘要：
设$L/K$是次数为$n=2m$的循环扩张。已知$L$上的交错$K$-双线性形式（斜形式）空间$\text{Alt}_K(L)$可分解为以$\text{Gal}(L/K)=\rangle\theta\theta$中元素为指标的$K$-子空间$A^{\rangle\theta^i\rangle}$的直和。同时已知分量$A^{\rangle\theta^i\rangle}$具有良好的常秩性质。我们改进并丰富了这些常秩结果，证明分量$A^\theta$通常可直和分解为若干常秩子空间之和，即这些子空间中所有非零斜形式都具有固定秩$r$。特别地，当$-1\not\rangle L^2$时这一结论恒成立。由此我们推导出在若干有意义的情形下$\text{Alt}_K(L)$可分解为常秩子空间之和。我们还证明，在$\rangle\theta^i\rangle$的阶可被$2$整除的$A^{\rangle\theta^i\rangle}$中，总能找到一个维数为$\frac{n}{2}$的子空间，其所有非零斜形式都是非退化的。
英文摘要：
Let $L/K$ be a cyclic extension of degree $n = 2m$. It is known that the space $\text{Alt}_K(L)$ of alternating $K$-bilinear forms (skew-forms) on $L$ decomposes into a direct sum of $K$-subspaces $A^{σ^i}$ indexed by the elements of $\text{Gal}(L/K) = \langle σ\rangle$. It is also known that the components $A^{σ^i}$ can have nice constant-rank properties. We enhance and enrich these constant-rank results and show that the component $A^σ$ often decomposes directly into a sum of constant rank subspaces, that is, subspaces all of whose non-zero skew-forms have a fixed rank $r$. In particular, this is always true when $-1 \not \in L^2$. As a result we deduce a decomposition of $\text{Alt}_K(L)$ into subspaces of constant rank in several interesting situations. We also establish that a subspace of dimension $\frac{n}{2}$ all of whose nonzero skew-forms are non-degenerate can always be found in $A^{σ^i}$ where $σ^i$ has order divisible by $2$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
