Set notation query

Question

What do square brackets mean next to sets? Like $\mathbb{Z}[\sqrt{-5}]$, for instance. I'm starting to assume it depends on context because google is of no use.

Answer 1

There is no general notation like that in set theory, that I'm aware of.

In your case, it's a notation from algebra, and it means: $\{m + n\sqrt{5}: m,n \in \mathbb{Z}\}$. We add a new number (here $\sqrt{5}$) to the integers and generate the minimal ring that contains them both.

Answer 2

it's the polynomial ring in the bracket with coefficients $\mathbf{Z}$, for example, $\mathbf{Z}[x]$ is the polynomial of x with coefficients $\mathbf{Z}$, like $x^3 + 2x^2 + 3$, and for $\mathbf{Z}[\sqrt{-5}]$, just replace x by $\sqrt-5$, the only difference is that $(\sqrt{-5}) ^2 = 5$, which is in $\mathbf{Z}$, but $x$ is independent of $\mathbf{Z}$
You can also find more formal definition in Lang's Algebra book(in the chapter of ring I think).