Is there any isomorphism between these two groups?
$(\mathbb{Q}^*,\cdot)$ and $(\mathbb{Z}_2,+)\times (\mathbb{Z}[x],+)$.
Is there any isomorphism between these two groups?
$(\mathbb{Q}^*,\cdot)$ and $(\mathbb{Z}_2,+)\times (\mathbb{Z}[x],+)$.
Hint. A nonzero rational number can be written as $(-1)^k\prod_{i=1}^{\infty}p_i^{a_i}$ where $p_1,p_2,\ldots$ are the distinct positive primes, and the $a_i$ are nonnegative integers, all but finitely many equal to $0$. The expression almost unique, two expressions only differing possibly in the value of $k$, but respecting its parity.
How is multiplication reflected in this way of expressing rationals?