TITLE: p-power roots of unity in local fields
QUESTION [8 upvotes]: Let $K$ be a number field and suppose $K$ contains no $p$-power roots of unity. Let $\mathcal{P}$ be a prime of $K$ above the rational prime $p$. Can someone prove or disprove the assertion that the local field $K_{\mathcal{P}}$ will contain no $p$-power roots of unity?

REPLY [6 votes]: Another counterexample, along the same lines as the one given by the other David S. but perhaps more "standard", is that $\mathbb{Q}_p(\zeta_p) = \mathbb{Q}_p((-p)^{1/(p-1)})$; so $K = \mathbb{Q}((-p)^{1/(p-1)})$ will do.  As "unknown" has mentioned, Krasner's lemma explains why you would expect this to be false.