Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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.

share|improve this question
add comment

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.)

share|improve this answer
    
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
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.