Every finite integral domain is field. I was wondering can some infinite integral domain be field?
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7Any infinite field is also an infinite integral domain – Gregory Simon May 12 '16 at 11:24
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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.
Dietrich Burde
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Every integral domain, including infinite ones, can be extended to an overring that is a field using localization. This construction is often called the quotient field or field of fractions. The integral domain $\mathbb{Z}$ has as its quotient field $\mathbb{Q}$. – hardmath May 12 '16 at 11:50