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.

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 , and an example of a real algebraic number is , both of which are of degree 2.

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

A number 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 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.

constant	degree
Conway's constant	71
Delian constant	3
disk covering problem	8
Freiman's constant	2
golden ratio	2
golden ratio conjugate	2
Graham's biggest little hexagon area	10
hard hexagon entropy constant	24
heptanacci constant	7
hexanacci constant	6
i	2
Lieb's square ice constant	2
logistic map 3-cycle onset	2
logistic map 4-cycle onset	2
logistic map 5-cycle onset	22
logistic map 6-cycle onset	40
logistic map 7-cycle onset	114
logistic map 8-cycle onset	12
logistic map 16-cycle onset	240
pentanacci constant	5
plastic constant	3
Pythagoras's constant	2
silver constant	3
silver ratio	2
tetranacci constant	4
Theodorus's constant	2
tribonacci constant	3
twenty-vertex entropy constant	2
Wallis's constant	3

If, 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).