what is $\mathbb{Z}[1/2]$?

Question

I have an exercise that uses the ring $\mathbb{Z}[1/2]$. I guess are integers but what $\frac{1}{2}$ means?

Answer 1

It's the ring of the numbers of the form $\frac n{2^m}$, with $n\in\mathbb Z$ and $m\in\mathbb Z^+$. More generally, if $\alpha\in\mathbb C$, then $\mathbb Z[\alpha]$ is the ring of the numbers of the form $a_0+a_1\alpha+\cdots+a_n\alpha^n$, with $a_0,a_1,\ldots,a_n\in\mathbb Z$.