I know this should be really easy, but I can't seem to figure it out. I don't really want an out an out answer if possible, just a starting point to work from.

The question reads as follows:

"Let R be an integral domain, and suppose $\displaystyle x \in R$ satisfies $\displaystyle x^2 = 1$. Prove that we must either have x = 1 or x = -1."

I started off by saying that if we have:

$\displaystyle x \cdot x = 1$

Then x has an inverse, $\displaystyle x^{-1}$ where $\displaystyle x^{-1} = x$.

This means that x is also in the group of units of R, but then I just don't know where to go... part of me wants to ignore the commutative law and say something like:

$\displaystyle x^2 = 1$

$\displaystyle x \cdot x = 1$

$\displaystyle x \cdot 1 = 1 \cdot x^{-1}$

So by right multiplication then x = 1 and $\displaystyle x^{-1} = 1$. But that just seems off.

Thanks for any help.

--

Dave