Ring of Gaussian integers
This article defines a particular commutative unital ring.
See all particular commutative unital rings
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 ).
|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|