# A Galois extension over $\mathbb{Q}$ with Galois group $A_4$ and with cyclic decomposition groups

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.

## 1 Answer

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 $3$.