Show that $(\mathbb Z[x],+)$ and $(\mathbb Q_{>0},\cdot)$ are isomorphic groups [closed]

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

Solution 1:

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.

1. Show that $(ℤ[x],+)$ is isomorphic to $(G,+).$ (Straightforward.)

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

Solution 2:

Hint: Use the fundamental theorem of arithmetic.

Solution 3:

What things additively generate $\Bbb Z[x]$? What do we always write polynomials as sums of?

What things multiplicatively generate $\Bbb Q_{>0}$? What are positive rationals always products (and quotients) of? If it helps, consider the question for natural numbers: what kinds of things do natural numbers always factor as products of?

See if you can use these hints to describe an isomorphism $(\Bbb Z[x],+)\cong(\Bbb Q_{>0},\cdot)$. If it helps, you can try to go through a middleman, $\Bbb Z^{\bigoplus\Bbb N}=\{(a_0,a_1,a_2,\dots):a_i\in\Bbb Z,\textrm{finitely many }a_i\ne0\}$.