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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 ?

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    $\begingroup$ Adjoin a suitable root of unity. $\endgroup$
    – LSpice
    Commented Mar 11, 2015 at 19:09
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    $\begingroup$ the $n$th cyclotomic field over $\mathbb{Q}$ is ramified at all primes dividing $n$. $\endgroup$
    – Will Chen
    Commented Mar 11, 2015 at 19:55
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    $\begingroup$ If $K$ is a local field, then there exists a unique unramified extension of each degree. See Serre's book on local fields. $\endgroup$
    – Daniel Loughran
    Commented Mar 11, 2015 at 20:58
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    $\begingroup$ See some previous MO questions: mathoverflow.net/questions/26491/… and mathoverflow.net/questions/41219/…. 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 $\mathbf Q$. $\endgroup$
    – KConrad
    Commented Mar 12, 2015 at 0:30

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