Rational Numbers under Addition form Infinite Abelian Group

Theorem
Let $\Q$ be the set of rational numbers.

The structure $\left({\Q, +}\right)$ is a countably infinite abelian group.

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

Hence by definition, $\left({\Q, +, \times}\right)$ is a field.

The fact that $\left({\Q, +}\right)$ forms an abelian group follows directly from the definition of a field.

From Rational Numbers are Countable, we have that $\left({\Q, +}\right)$ is countably infinite.