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Thread: Prove these two groups are isomorphic.

  1. #1
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    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
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  2. #2
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    Quote Originally Posted by whipflip15 View Post

    $\displaystyle \mbox{Prove the groups }(\mathbb{Z}[x],+)\mbox{ and }(\mathbb{Q}^+,*)\mbox{ are isomorphic.}$
    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.
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  3. #3
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    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 $\displaystyle f\in F$.
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  4. #4
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    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.
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