The Gaussian integers are the set
$${\displaystyle \mathbf {Z} [i]=\{a+bi\mid a,b\in \mathbf {Z} \},\qquad {\text{ where }}i^{2}=-1.}$$
In other words, a Gaussian integer is a complex number such … 展开

Since the ring G of Gaussian integers is a Euclidean domain, G is a principal ideal domain, which means that every ideal of G is principal. Explicitly, an ideal I is a subset of a ring R such … 展开

As for every unique factorization domain, every Gaussian integer may be factored as a product of a unit and Gaussian primes, and this factorization is unique up to the order of the factors, and the replacement of any prime by any of its associates (together with … 展开

As for any unique factorization domain, a greatest common divisor (gcd) of two Gaussian integers a, b is a Gaussian integer d that is a common divisor of a and b, which has all common divisors of a and b as divisor. That is (where | denotes the divisibility 展开

Gaussian integers have a Euclidean division (division with remainder) similar to that of integers and polynomials. This makes the Gaussian … 展开

As the Gaussian integers form a principal ideal domain they form also a unique factorization domain. This implies that a Gaussian integer is 展开

The field of Gaussian rationals is the field of fractions of the ring of Gaussian integers. It consists of the complex numbers whose real and … 展开