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17.52 (2010)

Open

(N. Eggert). Let $R$ be a commutative associative finite-dimensional nilpotent algebra over a finite field $F$ of characteristic $p$. Let $R^{(p)}$ be the subalgebra of all elements of the form $r^p$ ($r \in R$). Is it true that $\dim R \geqslant p \,\dim R^{(p)}$?

Contributor: L. S. Kazarin

Progress

An affirmative answer gives a solution to 11.44 for the case where the factors $A$ and $B$ are abelian.

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