How do you prove that a particular set is an integral domain? is it enough to prove that there are no zero divisors to say that it is not an integral domain?
For example: a + bsqrt(2) : a,b are integers. Would it be sufficient for me to say that since we can have [a + bsqrt(2)]*[0+0*sqrt(2)] = 0 our set cannot be an integral domain ?
Thank you,
On
The defining property of an integral domain is that we have the zero product law - i.e. if $ab=0$ then one of $a$ or $b$ is $0$. A statement like $$(a+b\sqrt{2})\cdot (0+0\sqrt{2})=0$$ does not contradict the zero product law, since $0+0\sqrt{2}=0$ (read this as "$0+0\sqrt{2}$ is the additive identity"), so the left hand does have a zero factor.
The ring $\mathbb{Z}[\sqrt{2}]$ is a subring of $\mathbb{C}$, which is a field, and hence has no nontrivial zero divisors. Hence $\mathbb{Z}[\sqrt{2}]$ itself is an integral domain.