On the Dimension of Nilpotent Algebras

The Eggert conjecture claims that a finite commutative algebra $R$ over a field of prime characteristic $p$ has the property $\dim R\ge p\dim R^{(1)}$, where $R^{(1)}$ is the subspace of $R$ spanned by the $p$th powers of elements of $R$. We obtain results related to this conjecture and results on nilpotent algebras of rather high nilpotency class.