Ring of Gaussian integers
This article is about a particular ring, i.e., a ring unique up to isomorphism. View a complete list of particular rings
Definition
The ring of Gaussian integers, denoted , is the set of complex numbers whose real and imagnary parts are both integers, under the usual addition and multiplication of complex numbers.
![{\displaystyle \mathbb {Z} [i]=\{a+bi:a,b\in \mathbb {Z} \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e7b83ae77e5f7f87eb2fc93e2498972c5b6ee9c2)
Ring properties
Basic properties
| Property | Satisfied? | Explanation |
|---|---|---|
| integral domain | Yes | |
| unique factorization domain | Yes | |
| principal ideal domain | Yes | |
| Euclidean domain | Yes | |
| field | No | No inverse of e.g. |
