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): Chapter $\text {IV}$: Rings and Fields: $23$. The Field of Rational Numbers: Theorem $23.8$