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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 maximal unramified extension $\mathbb{Q}_{2}^{\mathrm{unr}}$ 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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1 Answer 1

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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