Existence of Field of Quotients

Theorem
Let $\struct {D, +, \circ}$ be an integral domain.

Then there exists a field of quotients of $\struct {D, +, \circ}$.

Proof
Let $\struct {D, +, \circ}$ be an integral domain whose zero is $0_D$ and whose unity is $1_D$.

Inverse Completion is an Abelian Group
By Inverse Completion of Integral Domain Exists, we can define the inverse completion $\struct {K, \circ}$ of $\struct {D, \circ}$.

Thus $\struct {K, \circ}$ is a commutative semigroup such that:


 * $(1): \quad$ The identity of $\struct {K, \circ}$ is $1_D$


 * $(2): \quad$ Every element $x$ of $\struct {D^*, \circ}$ has an inverse $\dfrac {1_D} x$ in $\struct {K, \circ}$


 * $(3): \quad$ Every element of $\struct {K, \circ}$ is of the form $x \circ y^{-1}$ (which from the definition of division product, we can also denote $x / y$), where $x \in D, y \in D^*$.

It can also be noted that from Inverse Completion Less Zero of Integral Domain is Closed, $\struct {K^*, \circ}$ is closed.

Hence $\struct {K^*, \circ}$ is an abelian group.

Additive Operation on $K$
In what follows, we take for granted the rules of associativity, commutativity and distributivity of $+$ and $\circ$ in $D$.

We require to extend the operation $+$ on $D$ to an operation $+'$ on $K$, so that $\struct {K, +', \circ}$ is a field.

By Addition of Division Products, we define $+'$ as:


 * $\forall a, c \in D, \forall b, d \in D^*: \dfrac a b +' \dfrac c d = \dfrac {a \circ d + b \circ c} {b \circ d}$

where we have defined $\dfrac x y = x \circ y^{-1} = y^{-1} \circ x$ as $x$ divided by $y$.

Next, we see that:


 * $\forall a, b \in D: a +' b = \dfrac {a \circ 1_D + b \circ 1_D} {1_D \circ 1_D} = a + b$

So $+$ induces the given operation $+$ on its substructure $D$, and we are justified in using $+$ for both operations.

Addition on $K$ makes an Abelian Group
Now we verify that $\struct {K, +}$ is an abelian group

Taking the group axioms in turn:

$\text G 0$: Closure
Let $\dfrac a b, \dfrac c d \in K$.

Then $a, c \in D$ and $b, d \in D^*$, and $\dfrac a b + \dfrac c d = \dfrac {a \circ d + b \circ c} {b \circ d}$.

As $b, d \in D^*$ it follows that $b \circ d \in D^*$ because $D$ is an integral domain.

By the fact of closure of $+$ and $\circ$ in $D$, $a \circ d + b \circ c \in D$.

Hence $\dfrac a b + \dfrac c d \in K$ and $+$ is closed.

$\text G 1$: Associativity
Hence $\dfrac a b + \dfrac c d \in K$ and $+$ is associative.

$\text G 2$: Identity
The identity for $+$ is $\dfrac 0 k$ where $k \in D^*$:

Similarly for $\dfrac 0 k + \dfrac a b$.

$\text G 3$: Inverses
The inverse of $\dfrac a b$ for $+$ is $\dfrac {-a} b$:

From above, this is the identity for $+$.

Similarly, $\dfrac {-a} b + \dfrac a b = \dfrac 0 {b \circ b}$.

Hence $\dfrac {-a} b$ is the inverse of $\dfrac a b$ for $+$.

$\text C$: Commutativity
Therefore, $\struct {K, +, \circ}$ is a commutative ring with unity.

Product Distributes over Addition
From Extension Theorem for Distributive Operations, it follows directly that $\circ$ distributes over $+$.

Product Inverses in $K$
From Ring Product with Zero, we note that:
 * $\forall x \in D, y \in D^*: \dfrac x y \ne 0_D \implies x \ne 0_D$

From Inverse of Division Product:
 * $\forall x, y \in D^*: \paren {\dfrac x y}^{-1} = \dfrac y x$

Thus $\dfrac x y \in K$ has the ring product inverse $\dfrac y x \in K$.

Inverse Completion is a Field
We have that:
 * the algebraic structure $\struct {K, +}$ is an abelian group
 * the algebraic structure $\struct {K^*, \circ}$ is an abelian group
 * the operation $\circ$ distributes over $+$.

Hence $\struct {K, +, \circ}$ is a field.

We also have that $\struct {K, +, \circ}$ contains $\struct {D, +, \circ}$ algebraically such that:
 * $\forall x \in K: \exists z \in D, y \in D^*: z = \dfrac x y$

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