# Ring of Gaussian integers

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