Let G be the additive group if all polynomials in x with coefficients in $\displaystyle \mathbb{Z}$, and let H be the multiplicative group of all positive rationals. Prove that G $\displaystyle \cong$ H

I'm think if I list the prime numbers $\displaystyle p_0=2$, $\displaystyle p_1=3$, $\displaystyle p_2=5$,..., and define

$\displaystyle \varphi(e_0+e_1x+e_2x^2+...+e_nx^n)=p_0^{e_0}...p_ n^{e_n}$

But I'm really not sure where to go from here..