Existence of the \(N\)-th Root in Finite-Dimensional Power-Associative Algebras over Reals

Abstract

The paper is devoted to the solvability of equations in finite-dimensional power-associative algebras over \( \mathbb{R} \). Necessary and sufficient conditions for the existence of the \(n\)-th root in a power-associative \(\mathbb{R}\)-algebra are obtained. Sufficient solvability conditions for a specific class of polynomial equations in a power-associative \(\mathbb{R}\)-algebra are derived.