Symbol $p$-Algebras of Prime Degree and their $p$-Central Subspaces

Abstract

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 $d_1{s}_1+\cdots +d_{p+1}{s}_{p+1}$ for some non-negative integers $d_1,\dots ,d_{p+1}$ with $d_1+\cdots +d_{p+1}\le p-1$.