Dimensions of nilpotent algebras over fields of prime characteristic (Q1358975)

\textit{N. Eggert} [Pac. J. Math. 36, 631-634 (1971; Zbl 0213.32004)] conjectured that for a finite commutative nilpotent algebra \(M\) over a field \(K\) of characteristic \(p\) \(\dim_KM\geq p\dim_KM^{(p)}\) where \(M^{(p)}\) is the subalgebra of \(M\) generated by \(x^p\), \(x\in M\), and proved the conjecture in the case \(\dim_KM^{(p)}\leq 2\). The author slightly simplifies Eggert's proof and proves the conjecture for any finite dimensional nilpotent algebra \(M\) (not necessarily commutative) under the same condition \(\dim_KM^{(p)}\leq 2\).