Does there exist a Galois extension $L/\mathbb{Q}$ with Galois group $A_4$ (the alternating group on four letters) such that all the decomposition groups are cyclic?

This question is motivated by the answer by Kasper Andersen to my question. Namely, a desired example (if exists) would permit one to answer in the negative this hard question.


Yes. Note that Daniel Loughran's comment to David Speyer's answer to this question states that for any solvable group $G$, there is a Galois extension $L/\mathbb{Q}$ with all decomposition groups cyclic.

It's not hard to find a concrete example. If you let $L$ be the splitting field of $x^{4} - x^{3} - 7x^{2} + 2x + 9$, then $L/\mathbb{Q}$ is an $A_{4}$ extension ramified only at $163$. (This is one of the $A_{4}$ fields from Klueners and Malle's excellent database.) There are four prime ideals, $\mathfrak{p}_{1}$, $\mathfrak{p}_{2}$, $\mathfrak{p}_{3}$ and $\mathfrak{p}_{4}$ above $163$ in $L$, and so $ef = 3$ for each such prime ideal. Therefore, the decomposition group for each such prime ideal has order $3$.


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