Algebraic Integer

If $r$ is a Root of the Polynomial equation

$$x^n+a_{n-1}x^{n-1}+\ldots+a_1x+a_0=0,$$

where the $a_i$s are Integers and $r$ satisfies no similar equation of degree $<n$, then $r$ is an algebraic Integer of degree $n$. An algebraic Integer is a special case of an Algebraic Number, for which the leading Coefficient $a_n$ 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 $\geq 5$ 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 $\Bbb{Q}(\sqrt{-1}\,)$, since $a+bi$ are roots of

$$z^2-2az+a^2+b^2=0.$$

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.