Dimensions of A m {\displaystyle {\mathcal {A}}_{m}} , A m r {\displaystyle {\mathcal {A}}_{m}^{r}} and P m {\displaystyle {\mathcal {P}}_{m}}
dim ( A m r ) = dim ( A m ) − dim ( A m − 1 ) {\displaystyle \dim({\mathcal {A}}_{m}^{r})=\dim({\mathcal {A}}_{m})-\dim({\mathcal {A}}_{m-1})}
dim ( P m ) = dim ( A m ) − | { degree m products of lower degree primitive elements } | {\displaystyle \dim({\mathcal {P}}_{m})=\dim({\mathcal {A}}_{m})-|\{{\mbox{degree m products of lower degree primitive elements}}\}|}
