An algebraic number is a complex number 
that satisfies a nonzero polynomial
equation
 |
(1)
|
where 
and the 
are integers (or, equivalently, rational
numbers). Its degree is 
if it satisfies no such equation
of smaller polynomial degree.
A number that is not algebraic is said to be transcendental. An algebraic integer is an algebraic number satisfying a monic polynomial with integer coefficients.
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.
The conjugate elements of an algebraic number 
are the roots 
, 
, ..., 
of its algebraic
number minimal polynomial over 
. The corresponding monic polynomial
is
 |
(3)
|
Every polynomial with rational coefficients that vanishes at 
also vanishes at its conjugate
elements (Conway and Guy 1996).
The complex conjugate, real part, imaginary part, and complex
modulus of an algebraic number are also algebraic, although their degrees can
differ (Trott 2006). For a nonreal algebraic number 
of degree 
, the degree of 
is at most 
. To see this, form
 |
(4)
|
Its coefficients are symmetric polynomials in the conjugate elements,
so 
has rational coefficients.
Since 
is one of the other conjugate elements, 
. Thus 
is a polynomial of polynomial degree 
that vanishes at 
. For real 
, instead, 
has the same degree as 
.
The bound can be attained. For either nonreal root 
of 
, its complex modulus
has algebraic number minimal polynomial
 |
(5)
|
which is irreducible over 
and has polynomial degree
6.
An algebraic number approximating a numerical quantity can be found in the Wolfram Language using RootApproximant.
