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Written more formally, I have three (related questions):

Let $G$ be a finite group.

  1. Does there exist a Galois extension $K/\mathbb{Q}$ and a prime $P$ in $R := K \cap \mathbb{A}$ (the ring of algebraic integers in $K$) lying over $p$ s.t. $D(P|p) = G$? Here $D(P|p)$ refers to the decomposition group, i.e. $D(P|p) = \{\sigma \in \text{Gal}(K/\mathbb{Q}) : \sigma(P) = P\} $.

  2. Does there exist a Galois extension $K/\mathbb{Q}$ and a prime $P$ in $R := K \cap \mathbb{A}$ lying over $p$ s.t. $E(P|p) = G$? Here $D(P|p)$ refers to the inertia group, i.e. $E(P|p) = \{\sigma \in \text{Gal}(K/\mathbb{Q}) : \sigma(\alpha) \equiv \alpha \mod{P} \text{ for all }\alpha \in R\} $.

  3. If also given $H \trianglelefteq G$, does there exist a Galois extension $K/\mathbb{Q}$ and a prime $P$ in $R := K \cap \mathbb{A}$ lying over $p$ s.t. $E(P|p) = H$ and $D(P|p) = G$?

I thought of these problems while studying for an exam in algebraic number theory, and I have no idea if any of them are true, false or open.

I know that---in general---we don't know if we can solve the inverse Galois problem over $\mathbb{Q}$; none of these would (immediately) imply a solution to the inverse Galois problem, I'm pretty sure, and similarly the inverse Galois problem wouldn't imply any of these. These problems do feel somewhat close to the inverse Galois problem, as they require coming up with a Galois extension over $\mathbb{Q}$ with specific properties. I suppose I have a more general, somewhat vague question:

Is there any way to come up with examples (other than quadratics or cyclotomics) of number fields with certain properties?

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The answer to your first two questions is 'no' and the answer to the third one is 'yes', provided of course $G$ already appears as a decomposition group.

The key to all these is the fact that, given an extension of number fields $L/K$ and primes $P$ and $p$ with $P$ over $p$, the decomposition group is isomorphic to the Galois group of the completions $L_P/K_p$. In other words, every 'internal' question you want to answer about a decomposition group may just as well be a question about the Galois group of local field extensions.

So, for 1), it can be seen from the filtration of $G=Gal(L_P/K_p)=D(P)$ that $D(P)$ is solvable. Furthermore, there are many more restrictions placed on this group because this filtration gives an injection from $G_i/G_{i+1}$ (quotients of higher inertia groups) to quotients of the higher principal unit groups of $K$. The same reasoning puts constraints on the first inertia group.

Which solvable groups occur, then? Classifying them seems to be an open problem, see here: https://mathoverflow.net/questions/172569/local-inverse-galois-problem

For 3, the answer is yes by the local Galois correspondence.

Note that even classifying the abelian, rather than solvable, groups that occur as Galois groups of local fields is the whole point of local class field theory.

Finally, I find the last question too vague. It depends on what properties you mean.

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  • $\begingroup$ Thanks for the great answer; I had proven in an exercise that $D(P|p)$ is always solvable, but I completely forgot about that fact. $\endgroup$ – Marcus M Sep 20 '15 at 1:26

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