Is $(\mathbb{Q},+)$ the direct product of two non-trivial subgroups?

HINT: If $\Bbb Q=H\times G$, then there is an obvious homomorphism $h:\Bbb Q\to G$ with kernel $H$. Now consider this earlier question.

If $G$ is an abelian group and $G\simeq G_1\times G_2$, then there exist $H_1,H_2\le G$ two subgroups such that $G=H_1+H_2$ and $H_1\cap H_2=\{0\}$ (and viceversa!). Furthermore, $H_i$ can be chosen isomorphic to $G_i$, $i=1,2$.

If $H_i\le\mathbb Q$, $H_i\neq \{0\}$, then $H_i\cap\mathbb Z\neq \{0\}$ (why?), and therefore $(H_1\cap\mathbb Z)\cap(H_2\cap\mathbb Z)\neq \{0\}$. In particular, $H_1\cap H_2\neq\{0\}$.