Gaussian Integer

A Complex Number where and are Integers. The Gaussian integers are members of the Quadratic Field . The sum, difference, and product of two Gaussian integers are Gaussian integers, but only if there is an such that

Gaussian Integers can be uniquely factored in terms of other Gaussian Integers up to Powers of and rearrangements.

The norm of a Gaussian integer is defined by

Gaussian Primes are Gaussian integers for which is Prime and is a Prime Integer satisfying .
1. If , then and . These factors are equivalent since . For example, is not a Gaussian Prime.

2. If , then .

3. If , then or . If both do, then .

Every Gaussian integer is within of a multiple of a Gaussian integer .

See also Complex Number, Eisenstein Integer, Gaussian Prime, Integer, Octonion

References

Conway, J. H. and Guy, R. K. Gauss's Whole Numbers.'' In The Book of Numbers. New York: Springer-Verlag, pp. 217-223, 1996.

Shanks, D. Gaussian Integers and Two Applications.'' §50 in Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, pp. 149-151, 1993.

© 1996-9 Eric W. Weisstein
1999-05-25