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.]