Question as in the title, I need some help with how to prove
$\mathbb{Z}[p^{-1}]/\mathbb{Z}$ is a divisible abelian group.
Hints or answers are welcome. Thank you for your help.
How can I show $\mathbb{Z}[p^{-1}]/\mathbb{Z}$ is a divisible abelian group?
0
$\begingroup$
group-theory
homological-algebra
1 Answers
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An arbitrary element in $\mathbb{Z}[p^{-1}]/\mathbb{Z}$ may be written $\frac{a}{p^k}$ for some integers $k>0$ and $a The abelian group $\mathbb{Z}[p^{-1}]/\mathbb{Z}$ is divisible if for any integer $n>0$ we can write $\frac{a}{p^k}=n\frac{b}{p^l}$ for some $b,l$. First write $n=mp^i$ with $m$ not divisible by $p$. We have $m$ and $p^k$ coprime so there exist integers $x,y$ with $xm+yp^k=1$. Set $l=k+i$ and $b=ax$. This gives $n\frac{b}{p^l}=\frac{max}{p^k}=\frac{(1-yp^k)a}{p^k}=\frac{a}{p^k}$.