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I am not sure if the following questions have an answer.

(Question 1) Let $G$ be a finite Abelian group. Is it possible to find an unramified Abelian extension $L/K$ such that

$$G \cong \mathrm{Gal}(L/K)$$

Can we drop the condition of $G$ being Abelian?

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(Question 2) Let $G$ be a finite Abelian group. Is it possible to find an extensions $L/K$ such that

$$G \cong \mathrm{Gal}(L/K) \cong CL(K)$$

where $L$ is the Hilbert Class Field of $K$ and the last isomorphism follows from Artin reciprocity. This would give us a method to construct maximal unramified Abelian extensions.

Thanks in advance!

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  • $\begingroup$ The answer to the second question is surely unknown, see math.stackexchange.com/questions/10949/… $\endgroup$ – Ferra Apr 5 '14 at 15:26
  • $\begingroup$ If we COULD drop the condition of being abelian in Q1 surely the inverse galois problem would be solved? $\endgroup$ – fretty Apr 5 '14 at 18:19
  • $\begingroup$ @fretty: the inverse galois problem asks for extensions over the rationals (i.e. $K=\mathbb{Q}$). In my question this is not a requirement. $\endgroup$ – yannickvda Apr 5 '14 at 18:24
  • $\begingroup$ Well it is natural to ask for the groups which occur as extensions of an arbitrary number field. As far as I know this question is unsolved. $\endgroup$ – fretty Apr 5 '14 at 18:27
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The answer to the first question is yes. Construct an extension $K/F$ with given Galois group and lift it to an extension $KL/FL$ where $L$ and $K$ are disjoint and $L$ is chosen in such a way that it kills the ramification in $K/F$. This result was first proved by Arnold Scholz, then by Emil Artin in a private communication to Helmut Hasse (see their correspondence), and finally it was rediscovered by Ali Froehlich.

As for the second question, Ozaki has recently shown (Construction of maximal unramified p-extensions with prescribed Galois groups, Invent. Math. 183 )2011), 649-680) that every finite $p$-group is the Galois group of the $p$-class field tower of a suitable number field.

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