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Let $K \neq \mathbb{Q} $ be a finite extension of $\mathbb{Q}$. For a given integer $n$, how to construct an unramified extension of $K$ of degree $n$ ?

EDIT: If not then under what conditions on $K$, the above construction is possible ?

Suman
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    Adjoin a suitable root of unity. – LSpice Mar 11 '15 at 19:09
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    the $n$th cyclotomic field over $\mathbb{Q}$ is ramified at all primes dividing $n$. – Will Chen Mar 11 '15 at 19:55
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    If $K$ is a local field, then there exists a unique unramified extension of each degree. See Serre's book on local fields. – Daniel Loughran Mar 11 '15 at 20:58
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    See some previous MO questions: http://mathoverflow.net/questions/26491/is-there-a-ring-of-integers-except-for-z-such-that-every-extension-of-it-is-ram/26526#26526 and http://mathoverflow.net/questions/41219/number-fields-with-no-unramified-extensions. The first link shows $\mathbf Q(i)$ has no everywhere unramified extensions of degree greater than $1$, abelian or otherwise. So in that sense it is just like $\mathbf Q$. – KConrad Mar 12 '15 at 00:30

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