What does $\mathbb{Z}[\sqrt{-5}]$ mean? I'm reading What is the simplest non-principal ideal? and the notation $\mathbb{Z}[\sqrt{-5}]$ comes up.
After searching around on this site, I found Meaning of Z[X] set notation, which says $Z[X]$ means all polynomials with integer co-efficients.
But, that doesn't seem to make sense in the context of the linked post.
Can someone clarify what this notation means?
 A: Yes, it is all polynomials with integer coefficients with $\sqrt{-5}$ as the "variable". However, there's a simpler way to characterise the group...
If $x=\sqrt{-5}$, then $x^2=-5$ and $x^3=-5\sqrt{-5}$ and $x^4=25$, and so on.
Notice that for each $n\in\mathbb{N}$, $x^n$ is a number of the form $a+b\sqrt{-5}$, for some integers $a,b$. (We could be more specific at what types of $a,b$ we will get, but this is good enough.)
So, any polynomial in $x=\sqrt{-5}$ with integer coefficients will simplify down to an expression summing integer multiples of numbers of the shape $a+b\sqrt{-5}$, which itself will simplify down to a number of the shape $a+b\sqrt{-5}$.
For instance, $-x^3+3x+19=5\sqrt{-5}+3\sqrt{-5}+19=8\sqrt{-5}+19$.
So, $\mathbb Z[{\sqrt{-5}}]$ must be some subset of $\{a+b\sqrt{-5}:a,b\in\mathbb{Z}\}$.
Now, let's show that it is in fact the entire set $\{a+b\sqrt{-5}:a,b\in\mathbb{Z}\}$.
Pick some number of the form $a+b\sqrt{-5}$. This is in $\mathbb Z[{\sqrt{-5}}]$ because with $x=\sqrt{-5}$, it is equal to the linear term $bx+a$ (which is a polynomial).
So $\mathbb{Z}[\sqrt{-5}]=\{a+b\sqrt{-5}:a,b\in\mathbb{Z}\}$.
