# Group isomorphism between $(\mathbb{Z}[x], +)$ and $(\mathbb{Q}^+, \cdot)$ [duplicate]

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.

## marked as duplicate by Watson, Namaste abstract-algebra StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); Aug 19 '16 at 0:13

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.