Field of Quotients of Subdomain

Theorem
Let $$\left({F, +, \circ}\right)$$ be a field whose unity is $$1_F$$.

Let $$\left({D, +, \circ}\right)$$ be a subdomain of $$\left({F, +, \circ}\right)$$ whose unity is $$1_D$$.

If:

$$K = \left\{{\frac x y: x \in D, y \in D^*}\right\}$$

then $$\left({K, +, \circ}\right)$$ is a quotient field of $$\left({D, +, \circ}\right)$$.

Proof
$$1_D = 1_F$$ by Subdomain Test.

The sum and product of two elements of $$K$$ are also in $$K$$ by Addition of Division Products and Product of Division Products.

The additive and product inverses of $$K$$ are also in $$K$$ by Negative of Divided By and Inverse of Division Product.

Thus by Subfield Test, $$\left({K, +, \circ}\right)$$ is a subfield of $$\left({F, +, \circ}\right)$$ which clearly contains $$\left({D, +, \circ}\right)$$.

Hence the result.