关于交换域扩张中向量空间的乘积
On the product of vector spaces in a commutative field extension
摘要：
令$K \subset L$是可交换的域扩张。给定$L$的$K$ -子空间$A,B$，我们考虑积集$AB=\{ab \mid a \in A, b \in B\}$所张成的子空间$<AB>$.如果是$\dim_K A = r$和$\dim_K B = s$，那么$<AB>$的维数可以有多小？在本文中，我们在特征0中给出了这个问题的完整答案，并且更一般地针对可分扩展。在这种情况下，$\dim_K < AB>$的最佳下界由数值函数$$ κ_{K,L}(r,s) = \min_{h} (\lceil r/h\rceil + \lceil s/h\rceil -1)h, $$提供，其中$h$在所有有限维中间场$K \subset H \subset L$的$K$ -维的集合上运行。这个界限与加法数论中出现的一个界限密切相关。
英文摘要：
Let $K \subset L$ be a commutative field extension. Given $K$-subspaces $A,B$ of $L$, we consider the subspace $<AB>$ spanned by the product set $AB=\{ab \mid a \in A, b \in B\}$. If $\dim_K A = r$ and $\dim_K B = s$, how small can the dimension of $<AB>$ be? In this paper we give a complete answer to this question in characteristic 0, and more generally for separable extensions. The optimal lower bound on $\dim_K < AB>$ turns out, in this case, to be provided by the numerical function $$ κ_{K,L}(r,s) = \min_{h} (\lceil r/h\rceil + \lceil s/h\rceil -1)h, $$ where $h$ runs over the set of $K$-dimensions of all finite-dimensional intermediate fields $K \subset H \subset L$. This bound is closely related to one appearing in additive number theory.
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