Algebraic Numbers

Algebraic Number If is a root of a nonzero polynomial equation (1) where the s are integers (or equivalently, rational numbers) and satisfies no similar equation of degree , then is said to be an algebraic number of degree . A number that is not algebraic is said to be transcendental. If is an algebraic number and , then it is called an algebraic integer. In general, algebraic numbers are complex, but they may also be real. An example of a complex algebraic number is , and an example of a real algebraic number is , both of which are of degree 2. The set of algebraic numbers is denoted (Mathematica), or sometimes (Nesterenko 1999), and is implemented in Mathematica as Algebraics. A number can then be tested to see if it is algebraic in Mathematica using the command Element[x, Algebraics]. Algebraic numbers are represented in Mathematica as indexed polynomial roots by the symbol Root[f, n], where is a number from 1 to the degree of the polynomial (represented as a so-called "pure function") . Examples of some significant algebraic numbers and their degrees are summarized in the following table. constantdegreeConway's constant 71Delian constant 3disk covering problem 8Freiman's constant2golden ratio 2golden ratio conjugate 2Graham's biggest little hexagon area 10hard hexagon entropy constant 24heptanacci constant7hexanacci constant6i2Lieb's square ice constant2logistic map 3-cycle onset 2logistic map 4-cycle onset 2logistic map 5-cycle onset 22logistic map 6-cycle onset 40logistic map 7-cycle onset 114logistic map 8-cycle onset 12logistic map 16-cycle onset 240pentanacci constant5plastic constant3Pythagoras's constant 2silver constant3silver ratio2tetranacci constant4Theodorus's constant2tribonacci constant3twenty-vertex entropy constant2Wallis's constant3If, instead of being integers, the s in the above equation are algebraic numbers , then any root of (2) is an algebraic number. If is an algebraic number of degree satisfying the polynomial equation (3) then there are other algebraic numbers , , ... called the conjugates of . Furthermore, if 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, Euclidean Number, Hermite-Lindemann Theorem, Number Field, Radical Integer, Q-Bar, Transcendental Number