In number theory, 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]. The Gaussian integers are a special case of the quadratic integers. This domain does not have a total ordering that respects arithmetic.
Formally, Gaussian integers are the set
(Where the overline over "a+bi" refers to the complex conjugate.)
The norm is multiplicative, since the absolute value of complex numbers is multiplicative, i.e., one has
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 elements 1, −1, i and −i.
As a principal ideal domain
The Gaussian integers form a principal ideal domain with units 1, −1, i, and −i. If x is a Gaussian integer, the four numbers x, ix, −x, and −ix are called the associates of x. As for every principal ideal domain, the Gaussian integers form 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 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.
A Gaussian integer 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 (with n a nonnegative integer) or its negative , or
- both are nonzero and is a prime number (which will not be of the form ).
The following elaborates on these conditions.
The integer 2 factors as as a Gaussian integer, the second factorisation (in which i is a unit) showing that 2 is divisible by the square of a Gaussian prime; it is the unique prime number with this property.
The necessary conditions can be stated as following: if a Gaussian integer is a Gaussian prime, then either its norm is a prime number, or its norm is a square of a prime number. This is because for any Gaussian integer , notice
Here means “divides”; that is, if is a divisor of .
Now is an integer, and so can be factored as a product of prime numbers, by the fundamental theorem of arithmetic. By definition of prime element, if is a Gaussian prime, then it divides (in Z[i]) some . Also, divides
- , so in Z.
This gives only two options: either the norm of is a prime number, or the square of a prime number.
If in fact for some prime number , then both and divide . Neither can be a unit, and so
where is a unit. This is to say that either or , where .
However, not every prime number is a Gaussian prime. 2 is not because . Neither are prime numbers of the form because Fermat's theorem on sums of two squares assures us they can be written for integers and , and . The only type of prime numbers remaining are of the form .
Prime numbers of the form are also Gaussian primes. For suppose for , and it can be factored . Then . If the factorization is non-trivial, then . But no sum of squares of integers can be written . So the factorization must have been trivial and is a Gaussian prime.
If is a Gaussian integer whose norm is a prime number, then is a Gaussian prime, because the norm is multiplicative.
As an integral closure
As a Euclidean domain
It is easy to see graphically that every complex number is within units of a Gaussian integer.
Put another way, every complex number (and hence every Gaussian integer) has a maximal distance of
units to some multiple of z, where z is any Gaussian integer; this turns Z[i] into a Euclidean domain, where
The ring of Gaussian integers was introduced by Carl Friedrich Gauss in his second monograph on quartic reciprocity (1832) (see ). 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.
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?
- 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.
- Quadratic integer
- Hurwitz quaternion
- Eisenstein integer
- Algebraic integer
- Kummer ring
- Proofs of Fermat's theorem on sums of two squares
- Proofs of quadratic reciprocity
- Splitting of prime ideals in Galois extensions describes the structure of prime ideals in the Gaussian integers
- Table of Gaussian integer factorizations
- , OEIS sequence A002145 "COMMENT" section
- 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.
- 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.
- 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.
- www.alpertron.com.ar/GAUSSIAN.HTM is a Java applet that evaluates expressions containing Gaussian integers and factors them into Gaussian primes.
- www.alpertron.com.ar/GAUSSPR.HTM is a Java applet that features a graphical view of Gaussian primes.
- Henry G. Baker (1993) Complex Gaussian Integers for 'Gaussian Graphics', ACM SIGPLAN Notices, Vol. 28, Issue 11. DOI 10.1145/165564.165571 (html)
- IMO Compendium text on quadratic extensions and Gaussian Integers in problem solving
- Weisstein, Eric W., "Landau's Problems", MathWorld.