# Field of Quotients of Subdomain

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## Theorem

Let $\struct {F, +, \circ}$ be a field whose unity is $1_F$.

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

Let:

- $K = \set {\dfrac x y: x \in D, y \in D^*}$

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

Then $\struct {K, +, \circ}$ is a field of quotients of $\struct {D, +, \circ}$.

## 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, $\struct {K, +, \circ}$ is a subfield of $\struct {F, +, \circ}$ which clearly contains $\struct {D, +, \circ}$.

Hence the result.

$\blacksquare$

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 23$: Theorem $23.8$