Definition:Subfield

Subfield of Ring
Let $\left({R, +, \circ}\right)$ be a ring with unity.

Let $K$ be a subset of $R$ such that $\left({K, +, \circ}\right)$ is a field.

Then $\left({K, +, \circ}\right)$ is a subfield of $\left({R, +, \circ}\right)$.

Subfield of Field
The definition still holds for a field, by dint of the fact that a field is also a ring.

Let $\left({F, +, \circ}\right)$ be a field.

Let $K$ be a subset of $F$ such that $\left({K, +, \circ}\right)$ is also a field.

Then $\left({K, +, \circ}\right)$ is a subfield of $\left({F, +, \circ}\right)$.

Proper Subfield
Let $\left({K, +, \circ}\right)$ be a subfield of $\left({F, +, \circ}\right)$.

Then $\left({K, +, \circ}\right)$ is a proper subfield of $\left({F, +, \circ}\right)$ iff $K \ne F$.

That is, $\left({K, +, \circ}\right)$ is a proper subfield of $\left({F, +, \circ}\right)$ iff:
 * $\left({K, +, \circ}\right)$ is a subfield of $\left({F, +, \circ}\right)$;
 * $K$ is a proper subset of $F$.