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
are not uniquely factorable, since
Quadratic fields obey the identities
The Integers in the real field
are of the
form , where
There exist 22 quadratic fields in which there is a Euclidean Algorithm (Inkeri 1947).
See also Algebraic Integer, Eisenstein Integer, Gaussian Integer, Imaginary Quadratic Field, Integer, Number Field, Real Quadratic Field
Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, pp. 153-154, 1993.
© 1996-9 Eric W. Weisstein