The rationals as an additive group is free?

Question

Is there a set $X$ so that $(\mathbb{Q},+)$ are free on $X$ ?

Answer 1

Hint: take any two non-zero elements $a, b$ of $(\mathbb{Q},+)$. What can you say of $\langle a \rangle \cap \langle b \rangle$?

Answer 2

Try this:

Show that any pair of rational numbers are linearly dependent over $\mathbb{Z}$, so if it was free, it must have rank 1, i.e. be infinite cyclic. Then show that $\mathbb{Q}$ is not infinite cyclic.