Showing $2$ is not definable in $(\mathbb{Q},+)$.

Question

As stated, I'm to show that $2$ is not definable in $(\mathbb{Q},+)$.

I tried proving it by contradiction by showing that if $2$ were definable, then we could define $\mathbb{N}$ and multiplication over $\mathbb{N}$, which would be impossible because the automorphism $x\mapsto 2x$ does not preserve $\times$ over $\mathbb{N}$ but my definition of multiplication was flawed. Is there an easy trick to this problem?

Answer 1

Hint: Show that any definable element in a structure $M$ is fixed by all automorphisms of $M$. Can you find an automorphism of $\mathbb{Q}$ which does not fix $2$?