TITLE: Notation: $\mathbb{Z}[\sqrt{-5}]$
QUESTION [5 upvotes]: Show that the elements 2,3, and $1 \pm \sqrt{-5}$ are irreducible elements of $\mathbb{Z}[\sqrt{-5}]$.  I have never seen this notation before.  From another post I am interpreting this to mean the following:
$\mathbb{Z}[\sqrt{-5}] = \{a_{0}+a_{1}\sqrt{-5}+ \dots + a_{n}(\sqrt{-5})^{n} \colon a_{i} \in \mathbb{Z}     \}$.
Am I correct or is it something else?  I do not need the proof, just verification of notation.

REPLY [3 votes]: You are correct, but it it suffices to see that $\mathbb Z [\sqrt{-5}] = \{a_0 + a_1\sqrt{-5}: a_i \in \mathbb Z \}$