Prove that the additive group of $\mathbb{Z}[x]$ is isomorphic to the group of positive rational numbers under multiplication.

Question

The question (from my exam) quoted verbatim:

Show that the additive group of $\mathbb{Z}[x]$ is isomorphic to the group of positive rational numbers under multiplication.

First of all, is this $\mathbb{Z}[x]$ the same as $\mathbb{Z}_x$ (addition modulo $x$)?

Secondly, how do I still get the isomorphism?

Thanks

Answer 1

Hint: Enumerate the set of prime natural numbers with nonnegative integers. Say, $\{p_0,p_1,p_2,p_3,\ldots\}$ is such an enumeration. Prove that $$\sum_{n=0}^d\,k_n\,x^n\mapsto \prod_{n=0}^d\,p_n^{k_n}$$ is a group isomorphism from $\big(\mathbb{Z}[x],+\big)$ to $\big(\mathbb{Q}_{> 0},\cdot)$.