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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

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    $\begingroup$ $\Bbb Z[x]$ usually denotes the ring of univariate polynomial in the indeterminate $x$. Proceed by definition. $\endgroup$
    – xbh
    Commented Oct 3, 2018 at 4:49
  • $\begingroup$ Wait we haven't been taught rings, can't it be done from ONLY group theory? $\endgroup$
    – PLAP_
    Commented Oct 3, 2018 at 5:03
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    $\begingroup$ This is a choice of terminology. I could also just say that $\Bbb Z[x]$ is a set of univariate polynomials with integer coefficients, on which we could equip an addition operation. $\endgroup$
    – xbh
    Commented Oct 3, 2018 at 5:08

1 Answer 1

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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 $\big(\mathbb{Q}_{> 0},\cdot)$.

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  • $\begingroup$ First please explain to me what "exactly" is the set Z[x]? $\endgroup$
    – PLAP_
    Commented Oct 3, 2018 at 5:04
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    $\begingroup$ @PLAP_ Shouldn't have this been explained at some point in your course? It just means the set of polynomials with variable "x" with coefficients in $\mathbb{Z}$. See for example en.wikipedia.org/wiki/Polynomial_ring $\endgroup$
    – verret
    Commented Oct 3, 2018 at 5:36

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