Definition:Subfield

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 $\struct {F, +, \circ}$ $K \ne F$.

That is, $\struct {K, +, \circ}$ is a proper subfield of $\struct {F, +, \circ}$ :
 * $(1): \quad \struct {K, +, \circ}$ is a subfield of $\struct {F, +, \circ}$
 * $(2): \quad K$ is a proper subset of $F$.