Symbol p-algebras of prime degree and their p-central subspaces

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}).