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 $\struct {\Q, +, \times}$ of rational numbers is the field of quotients of the integral domain $\struct {\Z, +, \times}$ of integers.

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

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

Hence $\dfrac p p$ is the identity for $\struct {\Q, \times}$:

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

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