Exercise 3, § 13

Let K be an imperfect field and E an algebraic extension of K of finite degree. For an element $x \in E$ to exist such that $E = K(x)$ it is necessary and sufficient that the degree of imperfection of E over K (Ex. 1) should be 0 or 1. (To see that the condition is sufficient, note that if E, is the relative separable closure of K in E, we have $E = E_s(\alpha)$ and $E_s = K(\beta)$ and if $a \notin E_s$, consider the conjugates of $\alpha + c\beta$ over K, for $c \in K$.)