Exercise 1, § 4

Take $G =\mathbf{G}\mathbf{L}_n(k),n\geq 0$.

a) Show that $r_{Ad}^0(g) =n$ for all $g\in G$.

b) Show that an element $g\in G$ is regular if and only if its characteristic polynomial $P_g(T) =$ det(T$-g$) is separable; this is equivalent to saying that the discriminant (Algebra, Chap. IV, §1, no. 10) of $P_g(T)$ is $\not= 0$.