Gaussian integer

The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as

Gaussian integers are named after the German mathematician Carl Friedrich Gauss.

Since the Gaussian integers are closed under addition and multiplication, they form a commutative ring, which is a subring of the field of complex numbers.

The norm of a Gaussian integer is thus the square of its absolute value as a complex number.

This makes the Gaussian integers a Euclidean domain, and implies that Gaussian integers share with integers and polynomials many important properties such as the existence of a Euclidean algorithm for computing greatest common divisors, Bézout's identity, the principal ideal property, Euclid's lemma, the unique factorization theorem, and the Chinese remainder theorem, all of which can be proved using only Euclidean division.

In his original paper, Gauss made another choice, by choosing the unique associate such that the remainder of its division by 2 + 2i is one.

Multiplying g by the inverse of this unit, one finds an associate that has one as a remainder, when divided by 2 + 2i.

With the choices described above, the resulting unique factorization has the form where u is a unit (that is, u ∈ {1, –1, i, –i}), e0 and k are nonnegative integers, e1, …, ek are positive integers, and p1, …, pk are distinct Gaussian primes such that, depending on the choice of selected associates, An advantage of the second choice is that the selected associates behave well under products for Gaussian integers of odd norm.

On the other hand, the selected associate for the real Gaussian primes are negative integers.

The greatest common divisor of two Gaussian integers is not unique, but is defined up to the multiplication by a unit.

There are several ways for computing a greatest common divisor of two Gaussian integers a and b.

This process terminates, because, at each step, the norm of the second Gaussian integer decreases.

This method of computation works always, but is not as simple as for integers because Euclidean division is more complicated.

As a gaussian integer of norm 2 is necessary associated to 1 + i, and as 1 + i divides a and b, then the greatest common divisor is 1 + i.

In other words, two Gaussian integers are congruent modulo z0, if their difference belongs to the ideal generated by z0.

From this one can deduce by geometrical considerations, that the number of residue classes modulo a Gaussian integer z0 = a + bi equals its norm N(z0) = a2 + b2 (see below for a proof; similarly, for integers, the number of residue classes modulo n is its absolute value |n|).

This implies that all Qmn have the same area N = N(z0), and contain the same number ng of Gaussian integers.

Generally, the number of grid points (here the Gaussian integers) in an arbitrary square with the area A is A + Θ(√A) (see Big theta for the notation).

The residue class ring modulo a Gaussian integer z0 is a field if and only if

The number of its elements shall be denoted by ϕ(z) (analogously to Euler's totient function φ(n) for integers n).

For Gaussian primes it immediately follows that ϕ(p) = |p|2 − 1 and for arbitrary composite Gaussian integers Euler's product formula can be derived as where the product is to build over all prime divisors pm of z (with νm > 0).

Also the important theorem of Euler can be directly transferred: The ring of Gaussian integers was introduced by Carl Friedrich Gauss in his second monograph on quartic reciprocity (1832).

Gauss discovered that the law of biquadratic reciprocity and its supplements were more easily stated and proved as statements about "whole complex numbers" (i.e. the Gaussian integers) than they are as statements about ordinary whole numbers (i.e. the integers).

In a footnote he notes that the Eisenstein integers are the natural domain for stating and proving results on cubic reciprocity and indicates that similar extensions of the integers are the appropriate domains for studying higher reciprocity laws.

This paper not only introduced the Gaussian integers and proved they are a unique factorization domain, it also introduced the terms norm, unit, primary, and associate, which are now standard in algebraic number theory.

Most of the unsolved problems are related to distribution of Gaussian primes in the plane.