Ring of Gaussian integers

This article defines a particular commutative unital ring.
See all particular commutative unital rings

Definition

The ring of Gaussian integers $\mathbb{Z}[i]$ is defined in the following ways:

1. It is the subring generated by the ring of rational integers and the element $i$ (a square root of -1) in the field of complex numbers.
2. It is the integral extension $\mathbb{Z}[t]/(t^2+1)$ of the ring of rational integers $\mathbb{Z}$, with the image of the indeterminate $t$ denoted as $i$.
3. It is the ring of integers in the number field $\mathbb{Q}(i)$, a quadratic extension of the rationals given as $\mathbb{Q}[t]/(t^2+1)$ (with the image of $t$ denoted $i$).

Ring properties

Property	Meaning	Satisfied?	Explanation
integral domain	product of nonzero elements is nonzero	Yes	Follows from being a subring of $\mathbb{C}$
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 $\mathbb{C}$ from the closest point in $\mathbb{Z}[i]$ 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