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I'm looking for a group isomorphism between the group of integer polynomials (with addition) and the group of positive rationals (with multiplication).

I was thinking a map between generators of $\mathbb{Q}^*$, $1/p$, where $p$ is prime to irreducible polynomial in $\mathbb{Z}[x]$, but I am not exactly sure how it works. Any idea would be appreciated.

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marked as duplicate by Watson, Namaste abstract-algebra Aug 19 '16 at 0:13

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Hint: Define $$f:\Bbb{Z}[x]\to \Bbb Q^*:x^n\mapsto p_n$$ extended by linearity, where $p_n$ is the $n$th prime number. Show that $f$ is injective and surjective using the fundamental theorem of arithmetic.

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  • $\begingroup$ Missed the bit about positive rationals. :) Nice answer, +1. $\endgroup$ – Alex Wertheim Feb 19 '16 at 4:42

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