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 $1_D$.

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

where $\dfrac x y$ is the division product of $x$ by $y$.

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 Division Product 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.