# Definition:Algebraic Number

## Contents

- 1 Definition
- 2 Examples
- 2.1 Rational Number is Algebraic
- 2.2 $\sqrt 2$ is Algebraic
- 2.3 $\sqrt [3] {2 + \sqrt 2}$ is Algebraic
- 2.4 $\sqrt [3] 2 + \sqrt 3$ is Algebraic
- 2.5 $\sqrt 3 + \sqrt 2$ is Algebraic
- 2.6 $2 - \sqrt 2 i$ is Algebraic
- 2.7 $\sqrt [3] 4 - 2 i$ is Algebraic
- 2.8 Golden Mean is Algebraic
- 2.9 Imaginary Unit is Algebraic

- 3 Also see
- 4 Sources

## Definition

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

That is, it is a complex number that is a root 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 following:

### Rational Number is Algebraic

Let $r \in \Q$ be a rational number.

Then $r$ is also an algebraic number.

### $\sqrt 2$ is Algebraic

- $\sqrt 2$ is an algebraic element over the integers $\Z$.

### $\sqrt [3] {2 + \sqrt 2}$ is Algebraic

- $\sqrt [3] {2 + \sqrt 2}$ is an algebraic number.

### $\sqrt [3] 2 + \sqrt 3$ is Algebraic

- $\sqrt [3] 2 + \sqrt 3$ is an algebraic number.

### $\sqrt 3 + \sqrt 2$ is Algebraic

- $\sqrt 3 + \sqrt 2$ is an algebraic number.

### $2 - \sqrt 2 i$ is Algebraic

- $2 - \sqrt 2 i$ is an algebraic number.

### $\sqrt [3] 4 - 2 i$ is Algebraic

- $\sqrt [3] 4 - 2 i$ is an algebraic number.

### Golden Mean is Algebraic

- The golden mean $\phi = \dfrac {1 + \sqrt 5} 2$ is an algebraic number.

### Imaginary Unit is Algebraic

- The imaginary unit $i$ is an algebraic number.

## Also see

- Definition:Algebraic Integer
- Rational Number is Algebraic
- Definition:Algebraic Element of Field Extension

- Results about
**algebraic numbers**can be found here.

## Sources

- 1964: Steven A. Gaal:
*Point Set Topology*... (previous) ... (next): Introduction to Set Theory: $2$. Set Theoretical Equivalence and Denumerability - 1968: A.N. Kolmogorov and S.V. Fomin:
*Introductory Real Analysis*... (previous) ... (next): $\S 2$: Equivalence of Sets. The Power of a Set: Problem $4$ - 1981: Murray R. Spiegel:
*Theory and Problems of Complex Variables*(SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Solved Problems: Miscellaneous Problems: $47$ - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {B}.18$: Algebraic and Transcendental Numbers. $e$ is Transcendental