EGGERT’S CONJECTURE ON THE DIMENSIONS OF NILPOTENT ALGEBRAS

In this paper we prove that for a finite dimensional commu-tative nilpotent algebra A over a field of prime characteristic p> 0, dimA ≥ p dimA(p), where A(p) is the subalgebra of A generated by the elements xp. In particular, this solves Eggert’s conjecture. 1. Introduction. In 1971, Eggert [2] conjectured that for a finite commutative nilpotent al-gebra A over a field K of prime characteristic p> 0, dimA ≥ p dimA(p), where A(p) is the subalgebra of A generated by all the elements xp, x ∈ A and dimA, dimA(p) denote the dimensions of A and A(p) as vector space