# Definition:Subfield/Proper Subfield

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## Definition

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

- 1969: C.R.J. Clapham:
*Introduction to Abstract Algebra*... (previous) ... (next): Chapter $4$: Fields: $\S 16$. Subfields

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