Proof for chapter 17, H4

[wecing]

Ch17, 8.4: Prove "in any integral domain, only 1 and -1 are their own multiplicative inverses"

I don't quite get the solution; seems like the proof assumes "only 1 and -1 are their own multiplicative inverses". Please correct me if I am wrong! But I think I do have another proof:

Instead of solving a*a=1, we could let a=1+x, then solve (1+x)(1+x)=1, which expands to 1+x+x+x*x = 1, so x(1+1+x)=0. Integral domains do not have divisors of zero (theorem 2 on page 174), so at least one of x and 1+1+x must be zero (page 173), so either x=0 or x=-1-1; a=1 or a=-1.

[floation-cutie]

you're right! He didnt use the properity of integral domains.
We can also proof by contradiction.assume $t \ne -1$ and $t \ne 1$,use $t*t = 1$ to construct $(t-1)(t+1) = 0$
Same idea!