Rational $p$-adic integers and integers in $p$-adic field

Question

This is very basic question; if it is not suitable here, I will delete; but before deleting, it will be better, at least, if someone clarifies a little. In the book, Character Theory of Finite Groups by Huppert, the author take the subset of $\mathbb{Q}$ defined by $\mathbb{Z}_p=\{ \frac{a}{b} \,\,|a,b\in\mathbb{Z}, p\nmid b\}$, and calls it the ring of rational $p$-adic integers (p.$467$-$468$); is it the same (isomorphic) as the ring the one considered here?

Answer 1

No. They are different.

• The ring described in MathWorld is the ring of $p$-adic integers, $\Bbb{Z}_p$.
• That ring is an integral domain, and a completion of $\Bbb{Z}$ under the $p$-adic metric.
• The field of $p$-adic numbers, denoted $\Bbb{Q}_p$, is the field of fractions of $\Bbb{Z}_p$. Its elements have a similar description as power series of $p$, but this time we allow the lowest appearing exponent $m$ to also be a negative integer (so a finite point).
• We can view $\Bbb{Q}$ as a subset of $\Bbb{Q}_p$, and the ring $\Bbb{Z}_{(p)}$ Huppert describes is the intersection $\Bbb{Q}\cap\Bbb{Z}_p$, taken inside $\Bbb{Q}_p$.

Answer 2

Usually $\Bbb Z_p$ denotes the ring of $p$-adic integers, the completion of $\Bbb Z$ with respect to the $p$-adic topology. What Huppert is doing here is considering what I might write $\Bbb Z_{(p)}$, the localisation of $\Bbb Z$ with respect to the prime ideal $p\Bbb Z$. Then $\Bbb Z_p\not\cong\Bbb Z_{(p)}$, since the first is uncountable and the second is countable. But $\Bbb Q\cap\Bbb Z_p=\Bbb Z_{(p)}$ when $\Bbb Q$ and $\Bbb Z_p$ are considered as subrings of $\Bbb Q_p$ (the field of $p$-adic numbers). This, I suppose, justifies the use of the jargon rational $p$-adic integers.