Exercise 6, § 13

Let K be a commutative field and E an extension of K. Prove that the following properties are equivalent:

(i) For every separable extension L of K, the ring $E_{(L)}$ is semisimple.

(ii) The field E is a purely inseparable extension of a separable extension of K of finite degree.

(Assuming (i), use the corollary of V, §17, No. 2, p. 140 and Exercise 4 of V, §2, p. 146 to prove that E is an algebraic extension of K. Then, prove that the separable closure $E_s$ of K in E has finite degree. To do this, note that in the opposite case, the ring $E_{(L)}$, where L is a separable closure of K, would contain families of orthogonal idempotents $(e_1, . . . , e_n)$ with $n$ arbitrarily large. Assuming (ii), prove that if $E_{s(L)}$ is isomorphic to $\prod^r_{j=1}C_j$, then $E_{(L)}$ is isomorphic to $\prod^r_{j=1}D_j$, where $D_j$ is a purely inseparable extension of $C_j.$)