Idempotents in nonassociative algebras and eigenvectors of quadratic operators

Let $F$ be a field, char$(F)\mathrm{\ne }2$. Then every finite-dimensional $F$-algebra has either an idempotent or an absolute nilpotent if and only if over $F$ every polynomial of odd degree has a root in $F$. This is also necessary and sufficient for existence of eigenvectors for all quadratic operators in finite-dimensional spaces over $F$.