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](https://mathoverflow.net/q/241876/4149). Namely, a desired example (if exists) would permit one to answer in the negative [this hard question](https://mathoverflow.net/q/240439/4149).



A Galois extension over $\mathbb{Q}$ with Galois group $A_4$ and with cyclic decomposition groups
Yes. Note that Daniel Loughran's comment to David Speyer's answer to [this question](https://mathoverflow.net/questions/240439/hasse-principle-for-rational-times-square) 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](http://galoisdb.math.upb.de/groups/view?deg=4&num=4).) 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 $3$.


