# Definition:Algebraic Number

## Contents

- 1 Definition
- 2 Algebraic Number over Field
- 3 Examples
- 3.1 Rational Number is Algebraic
- 3.2 $\sqrt 2$ is Algebraic
- 3.3 $\sqrt {2 + \sqrt 3}$ is Algebraic
- 3.4 $\sqrt [3] {2 + \sqrt 2}$ is Algebraic
- 3.5 $\sqrt [3] 2 + \sqrt 3$ is Algebraic
- 3.6 $\sqrt 3 + \sqrt 2$ is Algebraic
- 3.7 $2 - \sqrt 2 i$ is Algebraic
- 3.8 $\sqrt [3] 4 - 2 i$ is Algebraic
- 3.9 Golden Mean is Algebraic
- 3.10 Imaginary Unit is Algebraic

- 4 Also see
- 5 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 degree of the minimal polynomial $P_n$ whose coefficients are all in $\Q$.

## Algebraic Number over Field

Some sources define an **algebraic number** over a more general field:

Let $F$ be a field.

Let $z$ be a complex number.

$z$ is **algebraic over $F$** if and only if $z$ is a root of a polynomial with coefficients in $F$.

## 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 number.

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

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

### $\sqrt [3] {2 + \sqrt 2}$ 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: $47$ - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next): Entry:**algebraic**:**3 a.** - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next): Entry:**algebraic number** - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {B}.18$: Algebraic and Transcendental Numbers. $e$ is Transcendental - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**algebraic number**