Dimensions of nilpotent algebras over fields of prime characteristic

Abstract

In this short paper we consider the conjecture that for a finite dimensional commutative nilpotent algebra M over a perfect field of prime characteristic p, dimM > pdimMŵ here M p is the subalgebra of M generated by x p ,x € M. We prove that for any finite dimensional nilpotent algebra M (not necessarily commutative) over any field of prime characteristic p, dim M > p dim M^ for dim M^ < 2.

Eggert's Conjecture for 2-Generated Nilpotent Algebras

“…Five years later, R. Bautista [2] proved it when dim A (p) = 3. C. Stack confirmed this results in Stack et al [3,4], but provided shorter proofs. Finally, Amberg and Kazarin [5] proved the conjecture for the case dim A (p) ≤ 4.…”

2-Generated nilpotent algebras and Eggert's conjecture

“…Five years later, R. Bautista [3] (1976) proved it when dim A (p) = 3. C. Stack confirmed this results in [10,11] (1996, 1998), but provided shorter proofs. Finally, B. Amberg and L.S.…”

Eggert’s conjecture on the dimensions of nilpotent algebras

“…In [3], Stack conjectures that dimA ≥ p dimA (p) is true for every finite dimensional nilpotent algebra A over K . We point out that some particular cases of Eggert's conjecture have been proved in [1,2,3,4].…”