An Algebraic Integer of the form where is Squarefree forms a quadratic field and is denoted
. If , the field is called a Real Quadratic Field, and if , it is called an
Imaginary Quadratic Field. The integers in
are simply called ``the''
Integers. The integers in
are called Gaussian Integers, and the integers in
are called Eisenstein Integers. The
Algebraic Integers in an arbitrary quadratic field do not necessarily have unique factorizations.
For example, the fields
and
are not uniquely factorable, since

(1) |

(2) |

Quadratic fields obey the identities

(3) |

(4) |

(5) |

The Integers in the real field
are of the
form , where

(6) |

There exist 22 quadratic fields in which there is a Euclidean Algorithm (Inkeri 1947).

**References**

Shanks, D. *Solved and Unsolved Problems in Number Theory, 4th ed.* New York: Chelsea, pp. 153-154, 1993.

© 1996-9

1999-05-25