Rational Addition Identity is Zero

Theorem
The identity of rational number addition is $0$:
 * $\exists 0 \in \Q: \forall a \in \Q: a + 0 = a = 0 + a$

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 Zero of Inverse Completion of Integral Domain, for any $k \in \Z^*$, the element $\dfrac {0_D} k$ of $\Q$ serves as the zero of $\left({\Q, +, \times}\right)$.

Hence $\dfrac 0 k$ is the identity for $\left({\Q, +}\right)$:

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

Next we note that it is a zero:

Hence we define the zero of $\left({\Q, +, \times}\right)$ as $0$ and identify it with the set of all elements of $\Q$ of the form $\dfrac 0 k$ where $k \in \Z^*$.