Proposition 14

— An endomorphism u of a finite dimensional vector space over a commutative field is semi-simple if and only if the minimal polynomial of u has no multiple factors.

This follows immediately from VII, p. 9, Cor. 4 and p. 31, Remark 1.

Let E be a vector space over a commutative field K, let L be an extension of K, and $u$ an endomorphism of E ; let $u_{(L)}$ denote the L-endomorphism $l_L \otimes u$ of the L-vector space $E_{(l.)} = L \otimes_K E$ induced from E by extension of scalars. In the same way, if $\mathfrak{F}$ is a set of endomorphisms of E, let $\mathfrak{F}_{(L)}$ denote the set of $u_{(L)}$ for $u$ in '8.