# Ring of Gaussian integers

From Commalg

This article defines a particular commutative unital ring.

See all particular commutative unital rings

## Definition

The **ring of Gaussian integers** is defined in the following ways:

- It is the subring generated by the ring of rational integers and the element (a square root of -1) in the field of complex numbers.
- It is the integral extension of the ring of rational integers , with the image of the indeterminate denoted as .
- It is the ring of integers in the number field , a quadratic extension of the rationals given as (with the image of denoted ).

## Ring properties

Property | Meaning | Satisfied? | Explanation |
---|---|---|---|

integral domain | product of nonzero elements is nonzero | Yes | Follows from being a subring of |

Euclidean domain | has a Euclidean norm | Yes | In fact, the standard algebraic norm (which in this case is the same as the square of the complex modulus) is a Euclidean norm. The key geometric fact used is that the distance of any point in from the closest point in is less than 1. |

principal ideal domain | integral domain and every ideal in it is a principal ideal | Yes | See Euclidean implies PID |

unique factorization domain | every element has a unique factorization into irreducibles up to units | Yes | See PID implies UFD |

Noetherian domain | integral domain | Yes | Follows from being a PID |

Bezout domain | Yes | Follows from being a PID | |

Dedekind domain | Yes | Follows from being a PID |