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.