Gaussian integer

The Gaussian numbers (after Carl Friedrich Gauss; engl Gaussian integer. ) Are a generalization of the integers in the complex numbers. Each figure is on a Gaussian integer coordinate point in the complex plane. The Gaussian numbers form the whole ring of quadratic field and in particular a ZPE ring and even a Euclidean ring. They occur for example in the formulation of the biquadratic reciprocity law. A somewhat more complex generalized integers, which may be embedded in the complex plane, the iron ore numbers.

Definition

A Gaussian number is by

Optionally, wherein and are integers.

The ring of Gaussian numbers is called also Gaussian speed ring and is denoted by. So he comes from by adjoining the imaginary unit.

The Gaussian numbers are the points with integer coordinates in the Gaussian number plane. They form a two-dimensional grid.

Prime elements

With the Gaussian numbers can be studied number theory. More particularly, prime elements can be defined as prime generalization of the notion. The uniqueness of the Primfaktordarstellung then also applies to the Gaussian numbers.

The primes in the ring of Gaussian numbers are closely related to ordinary prime numbers. They fall into three classes (up to multiplication by and, the units of the ring of Gaussian numbers):