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Algebraic Number


If r is a root of a nonzero polynomial equation

 a_nx^n+a_(n-1)x^(n-1)+...+a_1x+a_0=0,
(1)

where the a_is are integers (or equivalently, rational numbers) and r satisfies no similar equation of degree <n, then r is said to be an algebraic number of degree n.

A number that is not algebraic is said to be transcendental. If r is an algebraic number and a_n=1, then it is called an algebraic integer.

Any algebraic number is an algebraic period, and if a number is not an algebraic period, then it is a transcendental number (Waldschmidt 2006). Note there is a "gap" between those two statements in the sense that algebraic periods may be algebraic or transcendental.

In general, algebraic numbers are complex, but they may also be real. An example of a complex algebraic number is i, and an example of a real algebraic number is sqrt(2), both of which are of degree 2.

The set of algebraic numbers is denoted A (Wolfram Language), or sometimes Q^_ (Nesterenko 1999), and is implemented in the Wolfram Language as Algebraics.

A number x can then be tested to see if it is algebraic in the Wolfram Language using the command Element[x, Algebraics]. Algebraic numbers are represented in the Wolfram Language as indexed polynomial roots by the symbol Root[f, n], where n is a number from 1 to the degree of the polynomial (represented as a so-called "pure function") f.

Examples of some significant algebraic numbers and their degrees are summarized in the following table.

If, instead of being integers, the a_is in the above equation are algebraic numbers b_i, then any root of

 b_nx^n+b_(n-1)x^(n-1)+...+b_1x+b_0=0,
(2)

is an algebraic number.

If alpha is an algebraic number of degree n satisfying the polynomial equation

 (x-alpha)(x-beta)(x-gamma)...=0,
(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).


See also

Algebraic Integer, Algebraic Number Minimal Polynomial, Algebraic Number Theory, Algebraic Period, Euclidean Number, Hermite-Lindemann Theorem, Number Field, Radical Integer, Q-Bar, Transcendental Number Explore this topic in the MathWorld classroom

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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.Ferreirós, J. "The Emergence of Algebraic Number Theory." §3.3 in Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics. Basel, Switzerland: Birkhäuser, pp. 94-99, 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.Koch, H. Number Theory: Algebraic Numbers and Functions. Providence, RI: Amer. Math. Soc., 2000.Nagell, T. Introduction to Number Theory. New York: Wiley, p. 35, 1951.Narkiewicz, W. Elementary and Analytic Number Theory of Algebraic Numbers. Warsaw: Polish Scientific Publishers, 1974.Nesterenko, Yu. V. A Course on Algebraic Independence: Lectures at IHP 1999. Unpublished manuscript. 1999.Wagon, S. "Algebraic Numbers." §10.5 in Mathematica in Action. New York: W. H. Freeman, pp. 347-353, 1991.Waldschmidt, M. "Transcendence of Periods: The State of the Art." Pure Appl. Math. Quart. 2, 435-463, 2006.Wolfram, S. A New Kind of Science. Champaign, IL: Wolfram Media, p. 1168, 2002.

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Algebraic Number

Cite this as:

Weisstein, Eric W. "Algebraic Number." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/AlgebraicNumber.html

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