Rational Numbers under Addition form Infinite Abelian Group

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

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

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

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

Finally, we have that the Rational Numbers are Countable.