符号$p$ -素度代数及其$p$ -中心子空间
Symbol $p$-Algebras of Prime Degree and their $p$-Central Subspaces
摘要：
证明了素数度泛符号$p$ -代数$p$ -中心子空间的最大维数为$p+1$ .我们证明了如下数论事实：设$\{s_1,\dots,s_{p+1}\}$是加法群$G=(\mathbb{Z}/p \mathbb{Z}) \times (\mathbb{Z}/p \mathbb{Z})$中的$p+1$个不同的非零元素，则对于某些非负整数$d_1,\dots,d_{p+1}$，每个非零元素$g \in G$都可以表示为$d_1 s_1+\dots+d_{p+1} s_{p+1}$，其中$d_1+\dots+d_{p+1} \leq p-1$ .
英文摘要：
We prove that the maximal dimension of a $p$-central subspace of the generic symbol $p$-algebra of prime degree $p$ is $p+1$. We do it by proving the following number theoretic fact: let $\{s_1,\dots,s_{p+1}\}$ be $p+1$ distinct nonzero elements in the additive group $G=(\mathbb{Z}/p \mathbb{Z}) \times (\mathbb{Z}/p \mathbb{Z})$; then every nonzero element $g \in G$ can be expressed as $d_1 s_1+\dots+d_{p+1} s_{p+1}$ for some non-negative integers $d_1,\dots,d_{p+1}$ with $d_1+\dots+d_{p+1} \leq p-1$.
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