Let $\mathbb{Q}^{*}$ be the multiplicative group of positive rational numbers. Prove that $\mathbb{Q}^{*}$ is isomorphic to $(\mathbb{Z}[x],+)$.
My intuition is telling me that this is not true, but this is what my book is asking me to prove. I have tried to think of a isomorphic mapping for a while now and the best I can do is a homomorphism namely, $$\phi(a_0+a_1x+ \cdots a_nx^n)=\frac{a_0}{a_n}$$ If anyone could verify that these really aren't isomorphic or give me a hint on how to get started that would be greatly appreciated, thanks!
EDIT: From the hint in the comments I have come up with this, Let $a=p_1^{a_0}p_2^{a_1}\cdots p_n^{a_n}$ and $b=p_1^{b_0} p_2^{b_1} \cdots p_n^{b_n}$ so that $\frac{a}{b}=p_1^{a_0-b_0} \cdots p_n^{a_n-b_n}$ and then considering the mapping $$\phi(\frac{a}{b})=(a_0-b_0)+(a_1-b_1)x+ \cdots (a_n-b_n)x^n$$ then verifing that this is an isomorphism. Would this be correct?