Algebraic Integer

If  is a root of the polynomial equation

where the s are integers and  satisfies no similar equation of degree , then  is called 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 root extraction (the elementary operations) on rational numbers. In fact, if elementary operations are allowed on real numbers only, then there are real numbers which are algebraic integers of degree 3 that cannot be so expressed.

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

REFERENCES:

Ferreirós, J. "Algebraic Integers." §3.3.2 in Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics. Basel, Switzerland: Birkhäuser, pp. 97-99, 1999.

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.