Clarification on quadratic ring notation

Question

My Abstract Algebra text is using the notation $\mathbb{Z}[1 + \sqrt{-5}]$ and calling it a "quadratic integer ring." Just to clarify, $\mathbb{Z}[1 + \sqrt{-5}]$ is simply the set

$$ \left\{ a + b(1 + \sqrt{-5}) \ | \ a, b \in \mathbb{Z} \right\}, $$

correct?

Answer 1

Yes. In general, $\mathbb Z[\alpha]$ denotes the set of polynomial expressions in $\alpha$ (with integer coefficients), i.e., $$ \mathbb Z[\alpha]=\Bigl\{\,\sum_{k=0}^na_k\alpha^k\Bigm| n\in\mathbb N_0; a_0,\ldots,a_n\in\mathbb Z\,\Bigr\}.$$ In your case, $\alpha^2$ can be expressed with lower degrees, hence it is not necessary to go beyond elements of the form $a+b\alpha$.

Answer 2

It is correct. However, it can be written easier. We have

$$ \mathbb{Z}\left[1+\sqrt{-5}\right]=\{ a+b(1+\sqrt{-5})\mid a,b\in \mathbb{Z}\}=\mathbb{Z}[\sqrt{-5}]. $$ This is exactly the ring of integers in the quadratic number field $\mathbb{Q}(\sqrt{-5})$.

For integers not congruent $1$ modulo $4$ this is different. So, for example, the ring of interegs in the quadratic number field $\mathbb{Q}[\sqrt{-3}]$ is given by

$$ \mathbb{Z}\left[\frac{1+\sqrt{-3}}{2}\right]=\{ a+b(1+\sqrt{-3})/2\mid a,b\in \mathbb{Z}\}. $$