# Additive Group of Integers is Subgroup of Rationals

## Theorem

Let $\struct {\Z, +}$ be the additive group of integers.

Let $\struct {\Q, +}$ be the additive group of rational numbers.

Then $\struct {\Z, +}$ is a subgroup of $\struct {\Q, +}$.

## Proof

Recall that Integers form Integral Domain.

The set $\Q$ of rational numbers is defined as the field of quotients of the integers.

The fact that the integers are a subgroup of the rationals follows from the work done in proving the Existence of Field of Quotients from an integral domain.

$\blacksquare$