Roots of unity in $\mathbb{Q} _{11}$

Question

Here $\mathbb{Q} _{11}$ denotes the 11-adic field.

How can I show that the only root of unity of order 7 in this field is 1? Is it true that for any two distinct primes $p,q$, the only root of unity of order $q$ in $\mathbb{Q} _{p}$ is 1? (Or more generally - Is there some simple condition on $n\in \mathbb{N}$, $p$ prime, so that there are non-trivial roots of unity of order $n$ in $\mathbb{Q} _{p}$?)

Thanks.

Answer 1

Here is Proposition 15 from Chapter 3 of my lecture notes on a course on local fields.

Let $K$ be a Henselian [i.e., such that Hensel's Lemma applies: e.g. complete] discretely valued field of residue characteristic $p \geq 0$. Let $\mu'(K)$ denote the group of all roots of unity when $p = 0$ and the group of roots of unity of order prime to $p$ when $p > 0$. Then reduction modulo the maximal ideal induces an isomorphism from $\mu'(K)$ onto $\mu(k)$, the group of roots of unity of the residue field.

[The use of "Henselian" in the statement rather than the more traditional "complete" is partly to remind the reader that you should use Hensel's Lemma in the proof! Indeed, this is one of the first, most standard, easiest applications of HL.]

This applies in particular to show that the group of roots of unity in $\mathbb{Q}_{11}$ of order prime to $11$ is cyclic of order $10$, and thus there are no roots of unity of order $7$.

Later on in that section of my notes I show that $\mathbb{Q}_p$ does not have any nontrivial $p$-power roots of unity, and thus that its full group of roots of unit is cyclic of order $p-1$: one uses the (Schönemann-)Eisenstein criterion to see that the cyclotomic polynomial $\Phi_p$ is irreducible over $\mathbb{Q}_p$.

[All of this will be found in other sources which treat the arithmetic of local fields, of course.]

Answer 2

Exercise 1 (trivial): a root of unity is in the ring of integers of the field.

Exercise 2 (slightly less trivial, use the strong triangle inequality): the map from the ring of integers to the residue field $k$ given by reduction modulo the maximal ideal is injective on the roots of unity of order coprime to char $k$.

Exercise 3 (standard): the roots of unity in a finite field of order $q$ are precisely the ($q-1$)-st roots of unity.

Exercise 4: If $p$ is odd, $\mathbb{Q}_p$ has no $p$-th roots of unity.

Corollary: the roots of unity in $\mathbb{Q}_p$, $p$ odd, are the $(p-1)$-st roots of unity.

In particular, the answer to your $p$-$q$ question is "no". What is special about the pair 11 and 7 is that 7 is coprime to 11-1=10.