# Definition:Subfield

## Definition

Let $\struct {S, *, \circ}$ be an algebraic structure with $2$ operations.

Let $T$ be a subset of $S$ such that $\struct {T, *, \circ}$ is a field.

Then $\struct {T, *, \circ}$ is a **subfield** of $\struct {S, *, \circ}$.

The algebraic structure $\struct {S, *, \circ}$ is usually one of two types, as follows:

### Subfield of Ring

Let $\struct {R, +, \circ}$ be a ring with unity.

Let $K$ be a subset of $R$ such that $\struct {K, +, \circ}$ is a field.

Then $\struct {K, +, \circ}$ is a **subfield** of $\struct {R, +, \circ}$.

### Subfield of Field

The definition still holds for a field, by dint of the fact that a field is also a ring with unity.

Let $\struct {F, +, \circ}$ be a field.

Let $K$ be a subset of $F$ such that $\struct {K, +, \circ}$ is also a field.

Then $\struct {K, +, \circ}$ is a **subfield** of $\struct {F, +, \circ}$.

### Proper Subfield

Let $\struct {K, +, \circ}$ be a subfield of $\struct {F, +, \circ}$.

Then $\struct {K, +, \circ}$ is a **proper subfield** of $\struct {F, +, \circ}$ if and only if $K \ne F$.

That is, $\struct {K, +, \circ}$ is a **proper subfield** of $\struct {F, +, \circ}$ if and only if:

- $(1): \quad \struct {K, +, \circ}$ is a subfield of $\struct {F, +, \circ}$
- $(2): \quad K$ is a proper subset of $F$.

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 23$

- 1972: A.G. Howson:
*A Handbook of Terms used in Algebra and Analysis*... (previous) ... (next): $\S 6$: Rings and fields