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The set of Gaussian integer

$$ \mathbb{Z} [i] =\left\{ x + iy\mid x, y \in \mathbb{Z}\right\} $$

that is, those complex numbers whose real and imaginary parts are integer numbers, constitutes a so-called integer(or integrity) domain. I would like to know if that is somehow determined by the integer nature of their parts (when the domain is extended, such as to $\mathbb{Z}[\sqrt{-5}]$ , unique factorization collapses.

Thanks in advance.

drhab
  • 151,093
Javier Arias
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  • Any subring of an integral domain is an integral domain. Note that the Gaussian integers form a subring of $\mathbb C$. This is not the case for unique factorisation, where a subring of a UFD need not be a UFD. – Mathmo123 Apr 03 '15 at 14:18
  • @Mathmo123 Yes, I know, but does the fact of their being an integral domain have anything to do with their real and imaginary parts being integers? That is my question. – Javier Arias Apr 03 '15 at 14:25
  • Not particularly. For example $\mathbb Z[\sqrt 5]$ is also in integral domain. The name integral domain suggests that these rings are a generalisation the integers, capturing certain properties you'd expect from the integers. But an integral domain doesn't have to be made up of actual integers. – Mathmo123 Apr 03 '15 at 14:28
  • See here for more on the etymology of "integral domain". – Bill Dubuque Apr 03 '15 at 16:25

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