# Rational Numbers under Addition form Abelian Group

## Theorem

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

The structure $\struct {\Q, +}$ is a countably infinite abelian group.

## Proof

The rational numbers are, by definition, the quotient field of the integral domain $\struct {\Z, +, \times}$ of integers.

Hence by definition, $\struct {\Q, +, \times}$ is a field.

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

From Rational Numbers are Countably Infinite, we have that $\struct {\Q, +}$ is countably infinite.

$\blacksquare$