Can infinite integral domain be a field

Question

Every finite integral domain is field. I was wondering can some infinite integral domain be field?

Answer 1

Yes, $\mathbb{Q}$ is an infinite integral domain which is also a field (of course, any infinite field is also an infinite integral domain). What you may want to say is that not every infinite integral domain needs to be a field. Here $\mathbb{Z}$ is a good example, or more generally $\mathbb{Z}[\sqrt{d}]$ for $d\in \mathbb{Z}$.

Related: Every integral domain with finitely many ideals is a field.