Exercise 13, § 10

Let E be an algebraic extension of a field K, $E_s$ the largest separable extension and $E_r$ the largest p-radical extension of K contained in E. In an algebraic closure of K let N be the quasi-Galois extension of K generated by E ; then the largest separable extension of K contained in N is the quasi-Galois extension $N_s$ generated by $E_s$. For E, to be the largest p-radical extension of K contained in N, it is necessary and sufficient that E should be separable over $E_r$ (cf. V, p. 152, Ex. 3).