Corollary

— Let u be an endomorphism of a finite dimensional vector space over a commutative field K, and let L be an extension of K. If $u_{(L)}$ is semi-simple then u is semi-simple. If u is semi-simple and L is separable over K, then $u_{(L)}$ is semi-simple.

This follows immediately from Prop. 14 and from V, p. 120, Cor. 1 (note that the minimal polynomials of $u$ and $u_{(L)}$ coincide).