Terminology: How should we call $\mathbb{Z}[\sqrt{5}]$? I'm wondering, what shall we call the ring $\mathbb{Z}[\sqrt{5}]$? I know that $\mathbb{Z} \left[ \frac{1+\sqrt{5}}{2} \right]$ is called a quadratic integer ring. But do we have something similar for $\mathbb{Z}[\sqrt{5}]$? I'm sorry if the question seems a little bit dumb, but I haven't come across this term before. :(
Thank you very much,
And have a good day,
 A: Non-maximal subrings of rings of algebraic integers have been called "orders" (from German, I think) for a long time, although obviously this is not a wonderfully distinguishing choice, the word being used (in English) for many other purposes. Still, it is standard, so you can say "the order $\mathbb Z[\sqrt{5}]$ in the ring of algebraic integers $\mathbb Z[{1+\sqrt{5}\over 2}]$."
A: Rings of the form ${\mathbf Z}[\sqrt{d}]$ are not the most general type of order in a quadratic field. To distinguish them, a term that I like to use is pure quadratic ring, where "pure" refers to having a ring generator that is a pure square root $\sqrt{d}$ (as opposed to something like generator $(1+\sqrt{5})/2$). I call cubic orders of the form ${\mathbf Z}[\sqrt[3]{d}]$ pure cubic rings. Most cubic rings (= orders in cubic fields) are not generated by a pure cube root.
Of course, a pure cube root is the same thing as a cube root, and a pure square root is the same thing as a square root (of an integer). The label "pure" is just for emphasis.
I don't think this terminology is widespread. (Try googling "pure quadratic ring" or "pure cubic ring", and be sure to use quotes since otherwise, in the case of pure cubic ring, you're going to be swamped with web pages about rings made of pure cubic zirconium). But it feels natural, since the term "pure cubic field" for cubic fields of the form ${\mathbf Q}(\sqrt[3]{d})$ is common in number theory. See Henri Cohen's "A Course in Computational Algebraic Number Theory".
