# Definition:Smallest Field containing Subfield and Complex Number

## Definition

Let $F$ be a field.

Let $\theta \in \C$ be a complex number.

Let $S$ be the intersection of all fields such that:

- $S \subseteq F$
- $\theta \in F$

Then $S$ is denoted $\map F \theta$ and referred to as the **smallest field containing $F$ and $\theta$**.

### General Definition

Let $F$ be a field.

Let $\theta_1, \theta_2, \ldots, \theta_n \in \C$ be complex numbers.

Let $S$ be the intersection of all fields such that:

- $S \subseteq F$
- $\theta_1, \theta_2, \ldots, \theta_n \in F$

Then $S$ is denoted $\map F {\theta_1, \theta_2, \ldots, \theta_n}$ and referred to as the **smallest field containing $F$ and $\theta_1, \theta_2, \ldots, \theta_n$**.

## Examples

### Complex Numbers

The field of complex numbers is the smallest field containing $\R$ and $i$.

### Numbers of Type $a + b \sqrt 2: a, b \in \Q$

Let $\Q \sqbrk {\sqrt 2}$ denote the set:

- $\Q \sqbrk {\sqrt 2} := \set {a + b \sqrt 2: a, b \in \Q}$

that is, all numbers of the form $a + b \sqrt 2$ where $a$ and $b$ are rational numbers.

Then $\Q \sqbrk {\sqrt 2}$ is the smallest field containing $\Q$ and $\sqrt 2$.

### Smallest Field Containing $\Q$, $\sqrt 2$ and $\sqrt 3$

The smallest field containing $\Q$, $\sqrt 2$ and $\sqrt 3$ is:

- $\set {a + b \sqrt 2 + c \sqrt 3 + d \sqrt 6: a, b, c, d \in \Q}$

This forms a vector space of dimension $4$ which has basis $\set {1, \sqrt 2, \sqrt 3, \sqrt 6}$.

### Numbers of Type $a + b \sqrt [3] 2: a, b \in \Q$

Let $\Q \sqbrk {\sqrt [3] 2}$ denote the set:

- $\Q \sqbrk {\sqrt [3] 2} := \set {a + b \sqrt [3] 2 + c \sqrt [3] {2^2}: a, b, c \in \Q}$

Then:

- $\Q \sqbrk {\sqrt [3] 2}$ is the smallest field containing $\Q$ and $\sqrt [3] 2$

and:

- $\index {\Q \sqbrk {\sqrt [3] 2} } \Q = 3$

## Sources

- 1969: C.R.J. Clapham:
*Introduction to Abstract Algebra*... (previous) ... (next): Chapter $8$: Field Extensions: $\S 36$. The Degree of a Field Extension