Exercise 9.A.7

Answers

Proof. Suppose $ℕ_{C}$ is nilpotent. By 8.18, for any $v \in V$ , we have

0 + i 0 = (N_{}

C)^ $d i m V (v + i 0) = N^{\dim V} v + i 0$

where the second equality follows from 9.9. This implies that $N^{\dim V} v = 0$ . Thus $N$ is nilpotent.

The other direction is obvious from 9.9 and the definitions. □

celiopassos

2017-10-06 00:00