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 $$\frac {0_D} k$$ of $$\Q$$ serves as the zero of $$\left({\Q, +, \times}\right)$$.

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

$$ $$ $$

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

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 $$\frac 0 k$$ where $$k \in \Z^*$$.