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Encyclopedia > Gaussian integer

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 domain cannot be turned into an ordered ring, since it contains a square root of −1. In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ... The integers are commonly denoted by the above symbol. ... In abstract algebra, an integral domain is a commutative ring with an additive identity 0 and a multiplicative identity 1 such that 0 ≠ 1, in which the product of any two non-zero elements is always non-zero; that is, there are no zero divisors. ... Definitions In abstract algebra, an ordered ring is a commutative ring with a a total order such that if and , then if and , then . ...

Gaussian integers as lattice points in the complex plane
Gaussian integers as lattice points in the complex plane

Formally, Gaussian integers are the set Wikipedia does not have an article with this exact name. ... Wikipedia does not have an article with this exact name. ...

{a+bi mid a,bin mathbb{Z} }.

The norm of a Gaussian integer is the natural number defined as In linear algebra, functional analysis and related areas of mathematics, a norm is a function which assigns a positive length or size to all vectors in a vector space, other than the zero vector. ... In mathematics, a natural number is either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). The former definition is generally used in number theory, while the latter is preferred in set theory and computer science. ...

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 In mathematics, a unit in a ring R is an element u such that there is v in R with uv = vu = 1R. That is, u is an invertible element of the multiplicative monoid of R. The units of R form a group U(R) under multiplication, the group of...

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 In abstract algebra, an integral domain is a commutative ring with 0 ≠ 1 in which the product of any two non-zero elements is always non-zero. ... In mathematics, a prime number (or a prime) is a natural number that has exactly two (distinct) natural number divisors, which are 1 and the prime number itself. ... Modular arithmetic (sometimes called modulo arithmetic) is a system of arithmetic for integers, where numbers wrap around after they reach a certain value — the modulus. ... In mathematics, Pierre de Fermats theorem on sums of two squares states that an odd prime number p is expressible as with x and y integers, if and only if For example, the primes 5, 13, 17, 29, 37 and 41 are all congruent to 1 modulo 4, and...

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 and irreducible numbers are prime, since Z[i] is a Euclidean domain. So for example 2 + 3i is a Gaussian prime since its norm is 4 + 9 = 13.


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. Integral Closure of a Ring In abstract algebra, the concept of integral closure is a generalization of the set of all algebraic integers. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, the field of Gaussian rationals is the field Q(i) formed by adjoining the imaginary number i to the field of rationals. ... In mathematics, a rational number (commonly called a fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ...


It is easy to see graphically that every complex number is within frac{sqrt 2}{2} units of a Gaussian integer. Put another way, every complex number (and hence every Gaussian integer) is within frac{sqrt 2}{2}N(z) units of some multiple of z, where z is any Gaussian integer; this turns Z(i) into a Euclidean domain, where v(z) = N(z). In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ... In abstract algebra, a Euclidean domain (also called a Euclidean ring) is a type of ring in which the Euclidean algorithm can be used. ...

Contents

Historical background

The ring of Gaussian integers was introduced by Carl Friedrich Gauss in 1829 - 1831 (see [1]) while studying reciprocity laws which are generalisations of the theorem of quadratic reciprocity which he had first succeeded in proving in 1796. In particular, he was looking for relationships between p and q such that q should be a cubic residue of p (i.e. x^3equiv q ({rm mod} p)) or such that q should be a biquadratic residue of p (i.e. x^4equiv q ({rm mod} p)). During this research he discovered that some results were more easily provable by working in the ring of Gaussian integers, rather than the ordinary integers. (30 April 1777 – 23 February 1855) was a German mathematician and scientist of profound genius who contributed significantly to many fields, including number theory, analysis, differential geometry, geodesy, magnetism, astronomy and optics. ... In number theory, the law of quadratic reciprocity connects the solvability of two related quadratic equations in modular arithmetic. ...


He developed the properties of factorisation and proved the uniqueness of factorisation into primes in Z[i], and despite publishing little, he made some comments which indicate that he was aware of the significance of Eisenstein integers in stating and proving results on cubic reciprocity. Eisenstein integers as intersection points of a triangular lattice in the complex plane In mathematics, Eisenstein integers are complex numbers of the form aω + b where ω is a complex cube root of unity, and a and b are rational integers. ...


See also

Eisenstein integers as intersection points of a triangular lattice in the complex plane In mathematics, Eisenstein integers are complex numbers of the form aω + b where ω is a complex cube root of unity, and a and b are rational integers. ... Fermats theorem on sums of two squares states that an odd prime can be expressed as with and integers if and only if The only if clause is trivial: the squares modulo are and , so is congruent to , , or modulo . ... In mathematics, the interplay between the Galois group G of a Galois extension of number fields L/K, and the way the prime ideals P of the ring of integers OK factorise as products of prime ideals of OL, provides one of the richest parts of algebraic number theory. ...

Bibliography

  • C. F. Gauss, Theoria residuorum biquadraticorum. Commentatio secunda., Comm. Soc. Reg. Sci. Gottingen 7 (1832) 1-­34; reprinted in Werke, Georg Olms Verlag, Hildesheim, 1973, pp. 93-­148.
  • From Numbers to Rings: The Early History of Ring Theory, by Israel Kleiner (Elem. Math. 53 (1998) 18 – 35)

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