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?
Looks to me like this is false. Let $K = \mathbb{Q}(z)/(z^p1p^2)$. This is an extension of degree $p$, so it is disjoint from the pth cycloctomic field, and hence does not contain a $p$th root of $1$. Thus, it also can not contain a $p^k$th root of 1. Now, let's see how $z^p  1  p^2$ factors in Q_{p}. There is already one pth root of $1+p^2$ in Q_{p}; call this root a. (To see this, note that the power series $(1+x)^{1/p} = 1+(1/p)x + (1/\binom{p}{2})x^2 + ...$ converges for $x=p^2$.) Let $\mathcal{P}$ be a prime of $K$ corresponding to a factor of $z^p1p^2$ other than za. (In fact, $( z^p1p^2)/(za)$ is irreducible over $\mathbb{Q}_p$, but I don't need that.) So $K_\mathcal{P}$ contains a root b of $z^p1p^2$ other than $a$. But then $b/a$ is in $K_\mathcal{P}$ and is a $p$th root of 1. This might be true if you ask $K/\mathbb{Q}$ to be Galois, but I would bet against it. 


Another counterexample, along the same lines as the one given by the other David S. but perhaps more "standard", is that $\mathbb{Q}_p(\zeta_p) = \mathbb{Q}_p((p)^{1/(p1)})$; so $K = \mathbb{Q}((p)^{1/(p1)})$ will do. As "unknown" has mentioned, Krasner's lemma explains why you would expect this to be false. 


There is a more general heuristic here, which I bet the number theorists can state more precisely. Questions about Galois groups tend to be locally constant in the $p$adic setting. For example, consider the set of monic polynomials of degree $d$ with coefficients in $\mathbb{Q}_p$, topologized as $\mathbb{Q}_p^d$. Then I believe that properties such as "has a root in $\mathbb{Q}_p$", "splits completely in $\mathbb{Q}_p$", "has abelian Galois group over $\mathbb{Q}_p$" should be locally constant. This lead me to believe that I should be able to perturb $z^p1$ slightly to get a polynomial where the corresponding field still contained a root of $(z^p1)/(z1)$. 

