If you find any mistakes, please make a comment! Thank you.

In an integral domain, there are at most two square roots of 1

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.1 Exercise 7.1.11

Prove that if $R$ is an integral domain and $x^2 = 1$ for some $x \in R$, then $x = 1$ or $x = -1$.

Solution: If $x^2 = 1$, then $x^2 – 1 = 0$. Evidently, then, $$(x-1)(x+1) = 0.$$ Since $R$ is an integral domain, we must have $x-1 = 0$ or $x+1 = 0$; thus $x = 1$ or $x = -1$.


This website is supposed to help you study Linear Algebras. Please only read these solutions after thinking about the problems carefully. Do not just copy these solutions.
Close Menu