Corollary

— A necessary and sufficient condition for u to be absolutely semi-simple is that there exist an extension L of K such that L is perfect and $u_{(L)}$ is semi-simple.

The condition in the corollary means that there exists an extension L of K such that L is perfect and $q(X)$ has no multiple factors in $L[X]$ (Prop. 14) ; this condition is equivalent to (iii) by V, p. 38, Cor. 2.