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Is there any isomorphism between these two groups?

$(\mathbb{Q}^*,\cdot)$ and $(\mathbb{Z}_2,+)\times (\mathbb{Z}[x],+)$.

  • 1
    Could you please clarify what the second group is defined as? What does the $^*$ denote?2011-05-01
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    It is direct product and the former one refers to Q without zero.2011-05-01
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    Please use Mark-up, not boldface.2011-05-01

1 Answers 1

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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?