Prove a subring of rationals is a DVR

Question

I am currently stuck on a problem being:

Prove that the ring of rational numbers $\frac{a}{b}$ such that $p \nmid b$ for a prime integer p is a discrete valuation ring.

Any help would be appreciated.

Answer 1

We use the definition of discrete valuation ring in Atiyah and Macdonald.

The ring you mentioned is called $\mathbb{Z}_{(p)}$, and it has a map to $\mathbb{Z}_{\geq0}\cup\infty$ given by mapping a non-zero $\frac{a}{b}$ to $k$ where $k$ is the maximal interger such that $p^k|a$ and mapping $0$ to $\infty$.

Hence you can defined a discrete valuation on its fraction field $\mathbb{Q}$, namely $v(\frac{x}{y})=v(x)-v(y)$ where $x,y\in\mathbb{Z}_{(p)}-\{0\}$ and $v(0)=\infty$. You should check that this valuation is well-defined, and it is easy to see $\mathbb{Z}_{(p)}$ is the valuation ring of this discrete valuation.

Edit. In fact, the valuation over $\mathbb{Q}$ constructed above is exactly the $p$-adic valuation, and you get a norm $|x|_p=p^{-v_p(x)}$ over $\mathbb{Q}$. The only valuation over $\mathbb{Q}$ are $|\cdot|_p$ for some prime $p$ and the usual absolute value, this is proved in any book of algebraic number theory. The completion of $\mathbb{Q}$ associated to $p$-adic norm is the $p$-adic field $\mathbb{Q}_p$ and the completion of $\mathbb{Q}$ associated to is the usual absolute value is just $\mathbb{R}$. Studying the completion of number field at some prime is called $p$-adic number theory.