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Suppose $K$ is an extension of $\mathbb{Q}_{2}$ (I could ask this for any $p$, but I'm especially interested in $p = 2$) which contains the Hilbert Class Field (i.e., maximal unramified extension) of $\mathbb{Q}_{2}$ and also contains all roots of unity (including $2$-power ones!) Then does $K$ contain all roots of all elements of $\mathcal{O}_{K}^{\times}$?

This seems to boil down to knowing the ramification behavior of extensions obtained by adjoining roots of units in $\mathcal{O}_{K}^{\times}$. I feel like this should be hidden somewhere in the main results of class field theory, but I'm not sure.

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An example of a bad edit in my opinion. The HCF in the title is not clear. Local fields don't have Hilbert class fields, global fields have them. The version before wasn't good, but better. – Chris Wuthrich Jan 11 at 17:16
up vote 5 down vote accepted

Think of it this way : $K$ is an abelian extension of $\mathbf{Q}_p$. Now, the the extension $\mathbf{Q}_p(\root{p^m}\of u)$ need not even be galoisian over $\mathbf{Q}_p$ for some appropriate $u\in\mathbf{Z}_p^\times$, so it cannot be contained in $K$. (As it happens, $K$ is the maximal abelian extension of $\mathbf{Q}_p$ by the local Kronecker-Weber theorem, but this fact is not used in the above argument.)

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There you go, that argument should have occurred to me. I guess that although all the square roots are in $K$, higher roots generally are not. Thanks! – Jeff Yelton Jul 7 '13 at 18:39

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