Gaussian integer

From Free net encyclopedia

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.

Image:Gaussian integer lattice.png

Formally, Gaussian integers are the set

<math>\{a+bi | a,b\in \mathbb{Z} \}.</math>

The norm of a Gaussian integer is the natural number defined as

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

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

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. 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.

It is easy to see graphically that every complex number is within <math>\frac{\sqrt 2}{2}</math> units of a Gaussian integer. Put another way, every complex number (and hence every Gaussian integer) is within <math>\frac{\sqrt 2}{2}N(z)</math> 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).

See also

• Eisenstein integer
• Splitting of prime ideals in Galois extensions describes the structure of prime ideals in the Gaussian integers

External links

• http://www.alpertron.com.ar/GAUSSIAN.HTM is a Java applet that evaluates expressions containing Gaussian integers and factors them into Gaussian primes.
• http://www.alpertron.com.ar/GAUSSPR.HTM is a Java applet that features a graphical view of Gaussian primes.
• Gaussian Integers, Fermat's Last Theorem Blog traces the history of Fermat's Last Theorem from Diophantus of Alexandria to Andrew Wiles.
• [1] Complex Gaussian Integers for 'Gaussian Graphics'de:Gaußsche Zahl

fr:Entier de Gauss it:Intero gaussiano sl:Gaussovo praštevilo sv:Gaussiska primtal zh:高斯整數