# Thread: Prove these two groups are isomorphic.

1. ## Prove these two groups are isomorphic.

This question has me stumped and i think I am missing something obvious. Any help would be appreciated but i would rather a hint than a solution.

Question:
$\displaystyle \mbox{Prove the groups }(\mathbb{Z}[x],+)\mbox{ and }(\mathbb{Q}^+,*)\mbox{ are isomorphic.}$

Thanks

2. Originally Posted by whipflip15

$\displaystyle \mbox{Prove the groups }(\mathbb{Z}[x],+)\mbox{ and }(\mathbb{Q}^+,*)\mbox{ are isomorphic.}$
nice problem! let $\displaystyle p_n, \ n \geq 1,$ be the n-th prime number. define the map $\displaystyle \varphi: (\mathbb{Z}[x], +) \longrightarrow (\mathbb{Q}^{+}, \times)$ by: $\displaystyle \varphi(k_0 + k_1 x + \cdots + k_n x^n)=\prod_{j=0}^n p_{j+1}^{k_j}.$

so, for example: $\displaystyle \varphi(2+ 3x)=108, \ \varphi(1 -x + x^3)=\frac{14}{3},$ etc. i'm sure you can show that $\displaystyle \varphi$ is a group isomorphism.

3. I was thinking about the generalization of this problem. How about something to this effect. Let $\displaystyle D$ be a unique factorization domain and $\displaystyle F$ be a subgroup of $\displaystyle F(D)$ (the field of fractions) such that $\displaystyle f = u \prod _j \pi_j^{a_j}$ where $\displaystyle u$ if a fixed unit and $\displaystyle \pi_i$'s are from a particular representative set of irreducibles, with $\displaystyle a_j \in \mathbb{Z}$ for all $\displaystyle f\in F$.

4. The representative set would have to be countable if we are going to map the powers of x to the irreducibles. So D must also be infinite.