Exercise 4, § 13

Let K be a field of characteristic $p > 0$ and E an algebraic extension of K of finite degree. If $r > 0$ is the degree of imperfection of E over K (Ex. 1), show that $r$ is the least cardinal of sets S such that $E = K(S)$. (To see that E can be generated by $r$ of its elements note that if E, is the relative separable closure of K in E, then there exist r elements a, $(1 \leq i \leq r)$ of E such that $E = E_s(a_1, ..., a_r)$, and use Ex. 3.)