If is a Root of the Polynomial equation

where the s are Integers and satisfies no similar equation of degree , then is an algebraic Integer of degree . An algebraic Integer is a special case of an Algebraic Number, for which the leading Coefficient need not equal 1. Radical Integers are a subring of the Algebraic Integers.

A Sum or Product of algebraic integers is again an algebraic integer. However, Abel's Impossibility Theorem shows that there are algebraic integers of degree which are not expressible in terms of Addition, Subtraction, Multiplication, Division, and the extraction of Roots on Real Numbers.

The Gaussian Integers are algebraic integers of
, since are roots of

**References**

Hancock, H. *Foundations of the Theory of Algebraic Numbers, Vol. 1: Introduction to the General Theory.* New York: Macmillan, 1931.

Hancock, H. *Foundations of the Theory of Algebraic Numbers, Vol. 2: The General Theory.* New York: Macmillan, 1932.

Pohst, M. and Zassenhaus, H. *Algorithmic Algebraic Number Theory.* Cambridge, England: Cambridge University Press, 1989.

Wagon, S. ``Algebraic Numbers.'' §10.5 in *Mathematica in Action.* New York: W. H. Freeman, pp. 347-353, 1991.

© 1996-9

1999-05-25