Exercise 14, § 5

Let K be an imperfect field of characteristic p, let a be an element of $K - K^p$, and let L be the field $K(a^{1/p})$. Consider the K-vector space $E = L \otimes_L L$; let u (resp. u') be the K-endomorphism of E given by multiplication by $a^{1/p} \otimes 1$ (resp. $1 \otimes a^{1/p}$) in the algebra E. Show that u and u' are semi-simple, that u and $u'$ commute, and that $u - u'$ is nonzero nilpotent.