Proposition 15

— Let E be a finite dimensional vector space over a commutative field K, let u be an endomorphism of E and let q(X) be its minimal polynomial. Then the following conditions are equivalent :

(i) For every extension L of K, the endomorphism $u_{(L)}$ is semi-simple. (ii) There exists an extension L of K such that the endomorphism $u_{(L)}$ is diagonalisable. (iii) The polynomial q(X) is separable over K.

Indeed, condition (i) means that the polynomial $1 \otimes q(X)$ in $L[X]$ has no multiple factors, for any extension L of K (Prop. 14), condition (ii) means that there exists an extension L of K such that all the roots of $q(X)$ belong to L and that these roots are all simple (VII, p. 40, Prop. 12), and these conditions are each equivalent to (iii) by definition (V, p. 38).