# Definition:Algebraic Number

## Definition

An algebraic number is an algebraic element of the field extension $\C/\Q$.

That is, it is a complex number that is a roots of a polynomial with rational coefficients.

The set of algebraic number can often be seen denoted as $\mathbb A$.

### Degree

Let $\alpha$ be an algebraic number.

By definition, $\alpha$ is the root of at least one polynomial $P_n$ with rational coefficients.

The **degree** of $\alpha$ is the smallest $n$ such that $n$ is the degree of such a polynomial $P_n$.

## Examples

The algebraic numbers include:

- The elements of the set of rational numbers $\Q$, since these are roots of linear polynomials -- see Rational Number is Algebraic
- $\sqrt 2$, since it is a root of $x^2 - 2$
- $\sqrt [3] {2 + \sqrt 2}$, since it is a root of $x^6 - 4 x^3 + 2$
- $\sqrt 3 + \sqrt 2$, since it is a root of $x^4 - 10 x^2 + 1$
- $\sqrt [3] 4 - 2 i$, since it is a root of $x^6 + 12 x^4 - 8 x^3 + 48 x^2 + 96 x + 80$
- The golden section $\varphi$, since it is a root of $x^2 - x - 1$
- The imaginary unit $i$, since it is a root of $x^2 + 1$

## Also see

- Definition:Algebraic Integer
- Results about
**algebraic numbers**can be found here.

### Examples

### Generalizations

## Sources

- 1964: Steven A. Gaal:
*Point Set Topology*... (previous) ... (next): Introduction to Set Theory: $2$. Set Theoretical Equivalence and Denumerability - 1964: Murray R. Spiegel:
*Theory and Problems of Complex Variables*... (previous) ... (next): $1$: Solved Problems: $47$ - 1968: A.N. Kolmogorov and S.V. Fomin:
*Introductory Real Analysis*... (previous) ... (next): $\S 2$: Equivalence of Sets. The Power of a Set: Problem $4$ - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {B}.18$: Algebraic and Transcendental Numbers. $e$ is Transcendental