# Interpretation of a question: “group of all p-power roots of unity”

I have a homework problem I'm trying to do, but I'm not sure what it's asking. The problem is as follows:

Recall that $\mathbb{Q}/\mathbb{Z}$ is isomorphic to the group of all roots of unity in $\mathbb{C}$. Show that $\mathbb{Q}_p / \mathbb{Z}_p$ is isomorphic to the group of all $p$-power roots of unity. ($\mathbb{Z}_p$ is the p-adic integers, likewise $\mathbb{Q}_p$.)

What is the "group of all p-power roots of unity" in this context? Is it the union of the $p$-th roots of unity, $p^2$th roots of unity, $p^3$th roots of unity and so on? I don't think I've ever encountered the phrase before. (Presumably this is an obvious proof for some of you, and the answer is obvious, but for me I'm not so sure!)

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Yes. A "$p$-power root of unity" is a complex number $\zeta$ for which there exists a positive integer $n$ such that $$\zeta^{p^n}=1.$$That is, a complex number that is a $p^n$th root of unity for some positive integer $n$.
(More generally, an $m$th root of unity is a complex number that is a root of $x^m-1$, so we are just saying "a $p$th root of unity, or a $p^2$th root of unity, or a $p^3$th root of unity, or a ...")