I'm afraid this question might be considered opinion-based, but I dare. It might be considered opinion-based because it starts from a probably biased observation: In textbooks examples and discussions of simple extensions $\mathbb{Z}[\alpha]$ by adjunction of an irrational number $\alpha$ abound: $\sqrt{2}$,$\sqrt[3]{2}$, $\sqrt{-2},\dots$ But I rarely find thorough discussions of rational extensions like
$$\mathbb{Z}\Big[\frac{1}{2}\Big] = \Big\{a_0 + \frac{a_1}{2} + \dots + \frac{a_n}{2^n} \ |\ n \in \mathbb{N}, a_k \in \mathbb{Z}\Big\} = \Big\{\sum_{k=0}^n \frac{a_k}{2^k} \ |\ n \in \mathbb{N}, a_k \in \mathbb{Z}\Big\}$$
Is my impression deceptive, and extensions $\mathbb{Z}[\frac{1}{p}]$ with $p \in \mathbb{N}$ are in fact as important and interesing than extensions by irrational numbers? Or is it so, that they are not found so important and interesting? If so: For which reasons? If not so: In which contexts?
Why I find them interesting:
- Other than for square roots, the polyomial doesn't collapse:
$$\mathbb{Z}[\sqrt{p}] = \{a_0 + a_1\sqrt{p} + \dots + a_n\sqrt{p}^n \ |\ n \in \mathbb{N}, a_i \in \mathbb{Z}\} = \{a_0 + a_1\sqrt{p} \ |\ a_i \in \mathbb{Z}\}$$
$\mathbb{Z}[\frac{1}{p}] = \Big\{\sum_{k=0}^n \frac{a_k}{p^k} \ |\ n \in \mathbb{N}, a_k \in \mathbb{Z}\Big\} = \Big\{\sum_{k=0}^n \frac{a_k}{p^k} \ |\ n \in \mathbb{N}, |a_k| < p\Big\}$
Other than for square roots, each $x \in \mathbb{R}$ can be approximated arbitrary well in $\mathbb{Z}[1/p]$.
Nevertheless it's far from being $\mathbb{Q}$.
But it's not so far from being $\mathbb{R} = \Big\{\sum_{k=0}^\infty \frac{a_k}{p^k} \ |\ |a_k| < p\Big\}$ with its strong resemblance to the p-adic integers $\mathbb{Z}_p = \Big\{\sum_{k=0}^\infty a_k p^k \ |\ |a_k| < p\Big\}$
$\mathbb{Q} = \mathbb{Z}[\frac{1}{2}][\frac{1}{3}][\frac{1}{5}][\frac{1}{7}]\dots$ but you may not drop a single prime number
There will be more to say about $\mathbb{Z}[\frac{1}{p}]$ so any reference to a concise paper or textbook chapter would be very welcome.