**Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.1 Exercise 7.1.1**

Let $R$ be a ring with 1. Show that $(-1)^2 = 1$ in $R$.

Solution: Let $x \in R$. Then $$(-1)^2x = (-1)(-x) = x.$$ Similarly, $$x(-1)^2 = (-x)(-1) = x.$$ By the uniqueness of 1, we have $(-1)^2 = 1$.