A Gaussian integer is a complex number whose real and imaginary part are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as Z[i]. This is a Euclidean domain which cannot be turned into an ordered ring.
Formally, Gaussian integers are the set
The norm of a Gaussian integer is the natural number defined as
- N(a + bi) = a2 + b2.
The norm is multiplicative, i.e.
- N(z·w) = N(z)·N(w).
The units of Z[i] are therefore precisely those elements with norm 1, i.e. the elements
- 1, −1, i and −i.
The prime elements of Z[i] are also known as Gaussian primes. Some prime numbers (which, by contrast, are sometimes referred to as "rational primes") are not Gaussian primes; for example 2 = (1 + i)(1 − i) and 5 = (2 + i)(2 − i).
Those rational primes which are congruent to 3 (mod 4) are Gaussian primes; those which are congruent to 1 (mod 4) are not. This is because primes of the form 4k + 1 can always be written as the sum of two squares (Fermat's theorem), so we have
- p = a2 + b2 = (a + bi)(a − bi).
If the norm of a Gaussian integer z is a prime number, then z must be a Gaussian prime, since every non-trivial factorization of z would yield a non-trivial factorization of the norm. So for example 2 + 3i is a Gaussian prime since its norm is 4 + 9 = 13. This implies that since there are infinitely many ordinary primes then there must be infinitely many Gaussian primes.
The ring of Gaussian integers is the integral closure of Z in the field of Gaussian rationals Q(i) consisting of the complex numbers whose real and imaginary part are both rational.
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