TITLE: Show that $(\mathbb Z[x],+)$ and $(\mathbb Q_{>0},\cdot)$ are isomorphic groups
QUESTION [10 upvotes]: Let $(\mathbb{Z}[x],+)$ be the additive group of polynomials with integer coefficients and $ (\mathbb{Q}_{>0},\cdot)$ the multiplicative group of positive rationals. Show these groups are isomorphic.

Thanks

REPLY [10 votes]: One way of showing that two structures are isomorphic is to make a good choice of a third structure to which they're both isomorphic. To this end, let $G$ denote the set of all functions $f : P \rightarrow \mathbb{Z}$ (where $P = \{2,3,5,...\}$ is the set of all prime numbers) such that $f$ is zero for all but finitely many elements of its domain. Then $(G,+)$ is a group.
Exercises.

Show that $(ℤ[x],+)$ is isomorphic to $(G,+).$ (Straightforward.)
Show that $(\mathbb{Q}_{>0},\cdot)$ is isomorphic to $(G,+).$ Hint: Consider the function $\Pi : G \rightarrow \mathbb{Q}_{>0}$ defined by $\Pi(f) = \prod_{p \in P} p^{f(p)}.$