# Definition:Subfield

## Definition

### 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

- 1964: Iain T. Adamson:
*Introduction to Field Theory*... (previous) ... (next): $\S 1.2$ - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 23$ - 1969: C.R.J. Clapham:
*Introduction to Abstract Algebra*... (previous) ... (next): $\S 4.16$ - 1972: A.G. Howson:
*A Handbook of Terms used in Algebra and Analysis*... (previous) ... (next): $\S 6$: Rings and fields - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 56$: Definition $2$