Algebraic Number

If $r$ is a Root of the Polynomial equation

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

where the $a_i$s are Integers and $r$ satisfies no similar equation of degree $<n$, then $r$ is an algebraic number of degree $n$. If $r$ is an algebraic number and $a_0=1$, then it is called an Algebraic Integer. It is also true that if the $c_i$s in

$$c_0 x^n+c_1x^{n-1}+\ldots+c_{n-1}x+c_n=0$$ (2)

are algebraic numbers, then any Root of this equation is also an algebraic number.

If $\alpha$ is an algebraic number of degree $n$ satisfying the Polynomial

$$a(x-\alpha)(x-\beta)(x-\gamma)\cdots,$$ (3)

then there are $n-1$ other algebraic numbers $\beta$, $\gamma$, ... called the conjugates of $\alpha$. Furthermore, if $\alpha$ satisfies any other algebraic equation, then its conjugates also satisfy the same equation (Conway and Guy 1996).

Any number which is not algebraic is said to be Transcendental.

See also Algebraic Integer, Euclidean Number, Hermite-Lindemann Theorem, Radical Integer, Semialgebraic Number, Transcendental Number

References

Conway, J. H. and Guy, R. K. ``Algebraic Numbers.'' In The Book of Numbers. New York: Springer-Verlag, pp. 189-190, 1996.

Courant, R. and Robbins, H. ``Algebraic and Transcendental Numbers.'' §2.6 in What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 103-107, 1996.

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.

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