Additive group of all polynomials in x with integral coefficients is isomorphic to $\mathbb{Q}^*$

Question

Let G be the additive group of all polynomials in $x$ with integer coefficients. Show that G is isomorphic to the group $\mathbb{Q}$* of all positive rationals (under multiplication).

This question is from my abstract algebra assignment and I am unable to prove it. I am not able to deduce what should I map $a_0 +a_1 x + a_2 x^2 +...+ a_n x^n $ to so not able to proceed.

Can you please give a hint?

Answer 1

Hint

Consider the map

$$\begin{array}{l|rcl} \phi : & \mathbb Z[x] & \longrightarrow & \mathbb Q^* \\ & q(x)=q_0 + q_1 x + \dots + q_n x^n& \longmapsto & p_0^{q_0}p_1^{q_0} \cdots p_n^{q_n}\end{array}$$

where $p_i$ stands for the $i$-th prime number.

Answer 2

Hint: The fundamental theorem of arithmetic implies that $$ \mathbb{Q}^\times_+ \cong \bigoplus_p \mathbb Z $$