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 $F$.
Also see
- Results about subfields can be found here.
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $23$. The Field of Rational Numbers
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): subfield
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- 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 6$: Rings and fields