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In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as Z[i].[1] This integral domain is a particular case of a commutative ring of quadratic integers. It does not have a total ordering that respects arithmetic.
Formal definition

Formally, Gaussian integers are the set

$$\mathbf{Z}[i]=\{a+bi \mid a,b\in \mathbf{Z} \}, \qquad \text{ where } i^2 = -1.$$ [1]

Note that when they are considered within the complex plane the Gaussian integers may be seen to constitute the 2-dimensional integer lattice.
Norm of a Gaussian integer

The (arithmetic or field) norm of a Gaussian integer is the square of its absolute value (Euclidean norm) as a complex number. It is the natural number defined as

$$N(a+bi) = a^2+b^2 = (a+bi)\overline{(a+bi)} = (a+bi)(a-bi),$$

where  ⋅  (an overline) is complex conjugation.

The norm is multiplicative, since the absolute value of complex numbers is multiplicative, i.e., one has

N(zw) = N(z)N(w).[2]

The latter can also be verified by a straightforward check. The units of Z[i] are precisely those elements with norm 1, i.e. the set {±1, ±i}.[3]
As a principal ideal domain

The Gaussian integers form a principal ideal domain with units {±1, ±i}. For x ∈ Z[i], the four numbers ±x, ±ix are called the associates of x. As for every principal ideal domain, Z[i] is also a unique factorization domain.

The prime elements of Z[i] are also known as Gaussian primes. An associate of a Gaussian prime is also a Gaussian prime. The Gaussian primes are symmetric about the real and imaginary axes. The positive integer Gaussian primes are the prime numbers that are congruent to 3 modulo 4, (sequence A002145 in OEIS). One should not refer to only these numbers as "the Gaussian primes", which refers to all the Gaussian primes, many of which do not lie in Z.[4]

A Gaussian integer a + bi is a Gaussian prime if and only if either:

one of a, b is zero and the other is a prime number of the form 4n + 3 (with n a nonnegative integer) or its negative −(4n + 3), or
both are nonzero and a2 + b2 is a prime number (which will not be of the form 4n + 3).

In other words, a Gaussian integer is a Gaussian prime if and only if either its norm is a prime number, or it is the product by a unit (±1, ±i) of a prime number of the form 4n + 3.

It follows that there are three cases for the factorization of a prime number p in the Gaussian integers:

If p is congruent to 3 modulo 4, then it is a Gaussian prime; in the language of algebraic number theory, p is said to be inert in the Gaussian integers.
If p is congruent to 1 modulo 4, then it is the product of a Gaussian prime by its conjugate, both of which are non-associated Gaussian primes (neither is the product of the other by a unit); p is said to be a decomposed prime in the Gaussian integers.
If p = 2, we have 2 = (1 + i)(1 − i) = i(1 − i)2; that is, 2 is the product of the square of a Gaussian prime by a unit; it is the unique ramified prime in the Gaussian integers.

As an integral closure

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.
As a Euclidean domain

It is easy to see graphically that every complex number is within $$\tfrac{\sqrt 2}{2} units of a Gaussian integer. Put another way, every complex number (and hence every Gaussian integer) has a maximal distance of \( \frac{\sqrt 2}{2}\sqrt{N(z)}$$

units to some multiple of z, where z is any Gaussian integer; this turns Z[i] into a Euclidean domain, where

v(z) = N(z).[5]

Historical background

The ring of Gaussian integers was introduced by Carl Friedrich Gauss in his second monograph on quartic reciprocity (1832) (see [2]). The theorem of quadratic reciprocity (which he had first succeeded in proving in 1796) relates the solvability of the congruence x2 ≡ q (mod p) to that of x2 ≡ p (mod q). Similarly, cubic reciprocity relates the solvability of x3 ≡ q (mod p) to that of x3 ≡ p (mod q), and biquadratic (or quartic) reciprocity is a relation between x4 ≡ q (mod p) and x4 ≡ p (mod q). 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.

Unsolved problems
Repartition in the plane of the small Gaussian primes

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

Gauss's circle problem does not deal with the Gaussian integers per se, but instead asks for the number of lattice points inside a circle of a given radius centered at the origin. This is equivalent to determining the number of Gaussian integers with norm less than a given value.

There are also conjectures and unsolved problems about the Gaussian primes. Two of them are:

The real and imaginary axes have the infinite set of Gaussian primes 3, 7, 11, 19, ... and their associates. Are there any other lines that have infinitely many Gaussian primes on them? In particular, are there infinitely many Gaussian primes of the form 1+ki?[6]

Is it possible to walk to infinity using the Gaussian primes as stepping stones and taking steps of bounded length? This is known as the Gaussian moat problem; it was posed in 1962 by Basil Gordon and remains unsolved.[7][8]

Hurwitz quaternion
Eisenstein integer
Algebraic integer
Kummer ring
Proofs of Fermat's theorem on sums of two squares
Splitting of prime ideals in Galois extensions describes the structure of prime ideals in the Gaussian integers
Table of Gaussian integer factorizations

Notes

Fraleigh (1976, p. 286)
Fraleigh (1976, p. 289)
Fraleigh (1976, p. 288)
[1], OEIS sequence A002145 "COMMENT" section
Fraleigh (1976, p. 287)
Ribenboim, Ch.III.4.D Ch. 6.II, Ch. 6.IV (Hardy & Littlewood's conjecture E and F)
Gethner, Ellen; Wagon, Stan; Wick, Brian (1998). "A stroll through the Gaussian primes". The American Mathematical Monthly 105 (4): 327–337. doi:10.2307/2589708. MR 1614871. Zbl 0946.11002.

Guy, Richard K. (2004). Unsolved problems in number theory (3rd ed.). Springer-Verlag. pp. 55–57. ISBN 978-0-387-20860-2. Zbl 1058.11001.

References

C. F. Gauss, Theoria residuorum biquadraticorum. Commentatio secunda., Comm. Soc. Reg. Sci. Göttingen 7 (1832) 1-34; reprinted in Werke, Georg Olms Verlag, Hildesheim, 1973, pp. 93–148.
Fraleigh, John B. (1976), A First Course In Abstract Algebra (2nd ed.), Reading: Addison-Wesley, ISBN 0-201-01984-1
Kleiner, Israel (1998). "From Numbers to Rings: The Early History of Ring Theory". Elem. Math. 53 (1): 18–35. doi:10.1007/s000170050029. Zbl 0908.16001.
Ribenboim, Paulo (1996). The New Book of Prime Number Records (3rd ed.). New York: Springer. ISBN 0-387-94457-5. Zbl 0856.11001.