Definition:Subfield

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 $$F$$.