If is a Root of the Polynomial equation
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
See also Algebraic Number, Euclidean Number, Radical Integer
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.