Definition:Subfield

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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