Rational Multiplication Identity is One

Theorem
The identity of rational number multiplication is $$1$$:
 * $$\exists 1 \in \Q: \forall a \in \Q: a \times 1 = a = 1 \times a$$

Proof
From the definition, the field $$\left({\Q, +, \times}\right)$$ of rational numbers is the quotient field of the integral domain $$\left({\Z, +, \times}\right)$$ of integers.

From the properties of the quotient structure, elements of $$\Q$$ of the form $$\frac p p$$ where $$p \ne 0$$ act as the identity for multiplication.

From Equal Elements of Quotient Field, we have that:
 * $$\frac p p = \frac {1 \times p} {1 \times p} = \frac 1 1$$

Hence $$\frac p p$$ is the identity for $$\left({\Q, \times}\right)$$:

$$ $$

Similarly for $$\frac p p \times \frac a b$$.

Hence we define the unity of $$\left({\Q, +, \times}\right)$$ as $$1$$ and identify it with the set of all elements of $$\Q$$ of the form $$\frac p p$$ where $$ \in \Z^*$$.