How can I decide if $\mathbb{Q}_p$ contains the $\ell$-th roots of unity?

Question

I am trying to figure out how to come up with field extensions of $\mathbb{Q}_p$. Since abelian extensions are easy to write down over $\mathbb{Q}$, it seems natural to ask how to construct them for the p-adics. Given $\ell\neq p$ when is $$ \mathbb{Q}_p(\mu_\ell) $$ an abelian field extension?

Answer 1

The questions in your title and the body of your posting are different.

The question in the body is quite straightforward: $K(\mu_l)$ is an Abelian extension of $K$ for all fields $K$ and all $l$.

It is well-known that the roots of unity in $\Bbb Q_p$ are precisely the $(p-1)$-th roots if unity, when $p$ is odd, and the roots of unity in $\Bbb Q_2$ are $\pm1$.

Answer 2

As noted by Lord Shark the Unknown,"the questions in your title and the body of your posting are different". In interpret them as: 1) What are the roots of unity in $\mathbf Q_p$ ? 2) How to describe the abelian extensions of $\mathbf Q_p$ ?

1) To answer your first question, we may as well work in a $p$-adic local field $K$, i.e. a finite extension of $\mathbf Q_p$(throughout, take $p$ odd for simplification). Let $k$ be the residue field and $\pi$ an uniformizer of $K$. Write $v$ for the additive valuation of $k$, normalized by $v(\pi)=1$, so that $v(K^{*})=\mathbf Z$. The multiplicative group structure of $K^{*}$ is classically known : $K^{*} \cong (\mathbf Z,+) \times W \times U_1$, where $U_1 := 1 + (\pi)$ (the group of principal units) and $W$ is the group of roots of unity in $K$ of order prime to $p$ (by Hensel's lemma, $W$ is just a lift of $k^{*}$). If $K=\mathbf Q_p$, then $W \cong \mathbf F_p ^{*}$ and $U_1 = (1+p\mathbf Z_p) \cong (\mathbf Z_p,+)$ has no torsion, hence the roots of unity in $\mathbf Q_p$ are precisely the $(p-1)$-th roots of unity.

In the general case, it remains to look at $\mu_K$:= the group of $p$-power roots of unity in $K$, which is a priori contained in $U_1$. The degree of $K/\mathbf Q_p$ is $ef$ , where $f = [k:\mathbf F_p]$ is the inertia index, and the ramification index is defined as $e= v(p)$. The Galois theory of the cyclotomic extensions $\mathbf Q_p (\mu_{p^n})/\mathbf Q_p$ is exactly the same as over $\mathbf Q$ (Serre, "Local Fields", end of chap.IV), so a necessary condition for $\mu \neq (1)$ is that $(p-1)$ divides $e$. To go further, one must study the effect of raising to the $p$-th power in $U_1$, and this depends on $K$ .

2) I guess that your second question is about some kind of "local Kronecker-Weber" theorem over $\mathbf Q_p$. This indeed exists and is very easy to formulate. Take $K = \mathbf Q_p$ and let $K_{p^{\infty}}$ the infinite extension obtained by adding all the $p$-power roots of unity. Similarly, let $K_{nr}$ be the infinite extension obtained by adding all the roots of unity of order prime to $p$ (this is known to coincide with the maximal unramified extension of $K$, hence the notation). Local CFT (independently of the global K-W theorem) allows to show that the maximal abelian extension of $\mathbf Q_p$ is no other than the compositum $K_{nr}.K_{p^{\infty}}$ (op. cit., chap. XIV). A direct proof can be found in chap. 14 of Washington's "Introduction to cyclotomic fields" .