Definition:Subfield/Proper Subfield

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
• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): proper: 2.