Exercise 3, § 1

We make the assumptions in no. 2. Assume that $k$ is infinite and that condition (AC) is satisfied. Let $k'$ be a perfect extension of $k$. Let $\lambda : S\rightarrow k$ be such that $V^{\lambda}(S)\not= 0$. Put

Let $r': S'\rightarrow$ End(V$'$) be the linear map obtained from $r$ by extension of scalars. There exists a unique map $\lambda ': S'\rightarrow k'$ such that $V^{\lambda}(S)\otimes_kk'=$ ${V'}^{\lambda'}(S')$. (Reduce to the case where $V = V^{\lambda}(S)$. Let P be a polynomial function on S and $q$ a power of the characteristic exponent of $k$ dividing dim V, such that $\lambda^q= P$. Let $P'$ be the polynomial function on $S'$ which extends P. For each $s'\in S'$, there exists a $\lambda '(s')\in k'$ such that $\lambda '(s')^q= P'(s')$. Show that the characteristic polynomial of $r'(s')$ is $(X-\lambda '(s'))^{dim V}.$)