**Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.32**

Solution: Suppose $x^n = 0$. Then $$\varphi(x)^n = \varphi(x^n) = \varphi(0) = 0,$$ so that $\varphi(x)$ is nilpotent.

**Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.32**

Solution: Suppose $x^n = 0$. Then $$\varphi(x)^n = \varphi(x^n) = \varphi(0) = 0,$$ so that $\varphi(x)$ is nilpotent.