# 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 $S'$ such that:

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

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

This page needs the help of a knowledgeable authority.In particular: I encounter this definition in 1969: C.R.J. Clapham: Introduction to Abstract Algebra without having much idea of the context. The element $\theta$ is specifically defined as being a complex number. At the moment I have not got a clue as to why the specific nature of $\theta$, but I will go along with it as defined in Clapham. If the context needs to be expanded, that can happen in due course.If you are knowledgeable in this area, then you can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by resolving the issues.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Help}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

### 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 $S'$ such that:

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

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