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Gaussian integer

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. This integral domain is a particular case of a commutative ring of quadratic integers. It does not have a total ordering that respects arithmetic.The Gaussian integers are the setGaussian integers have a Euclidean division (division with remainder) similar to that of integers and polynomials. 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.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 that every sum of elements of I and every product of an element of I by an element of R belong to I. An ideal is principal, if it consists of all multiples of a single element g, that is, it has the formAs the Gaussian integers form a principal ideal domain they form also a unique factorization domain. This implies that a Gaussian integer is irreducible (that is, it is not the product of two non-units) if and only if it is prime (that is, it generates a prime ideal).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 a corresponding change of the unit factor).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 imaginary part are both rational.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 relation),Given a Gaussian integer z0, called a modulus, two Gaussian integers z1,z2 are congruent modulo z0, if their difference is a multiple of z0, that is if there exists a Gaussian integer q such that z1 − z2 = qz0. In other words, two Gaussian integers are congruent modulo z0, if their difference belongs to the ideal generated by z0. This is denoted as z1 ≡ z2 (mod z0).The relation Qmn = (m + in)z0 + Q00 means that all Qmn are obtained from Q00 by translating it by a Gaussian integer. This implies that all Qmn have the same area N = N(z0), and contain the same number ng of Gaussian integers.Many theorems (and their proofs) for moduli of integers can be directly transferred to moduli of Gaussian integers, if one replaces the absolute value of the modulus by the norm. This holds especially for the primitive residue class group (also called multiplicative group of integers modulo n) and Euler's totient function. The primitive residue class group of a modulus z is defined as the subset of its residue classes, which contains all residue classes a that are coprime to z, i.e. (a,z) = 1. Obviously, this system builds a multiplicative group. The number of its elements shall be denoted by ϕ(z) (analogously to Euler's totient function φ(n) for integers n).The ring of Gaussian integers was introduced by Carl Friedrich Gauss in his second monograph on quartic reciprocity (1832). 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).Most of the unsolved problems are related to distribution of Gaussian primes in the plane.

[ "Integer", "Combinatorics", "Discrete mathematics", "Algebra", "Cubic reciprocity", "Gaussian rational" ]
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