Leopoldt-type theorems for non-abelian extensions of \(\mathbb{Q}\) (Q6634402)

Let \(K\) be a normal algebraic number field with Galois group \(G = \text{Gal}(K/\mathbb Q)\). Then the ring of integers \(\mathcal O_K\) of \(K\) is a module over the associated order\N\[\N\mathfrak A _{K/\mathbb Q} = \{ \lambda \in \mathbb Q [G] \mid \lambda \mathcal O_K \subseteq \mathcal O_K \}.\N\]\NOne is interested in the question whether \(\mathcal O_K\) is free over \(\mathfrak A _{K/\mathbb Q}\), which is known to be true for the case that \(G\) is abelian or dihedral of order \(2p\) with a prime \(p\), or \(G\) is isomorphic to the quaternion group \(Q_8\) and \(K/\mathbb Q\) is wildly ramified.\N\NThe author focusses on the situation where the locally free class group \(\text{Cl}(\mathbb Z [G])\) is trivial, and therefore it suffices to investigate whether \(\mathcal O_K\) is locally free over \(\mathfrak A _{K/\mathbb Q}\). Apart from the case of dihedral groups, the cases of \(G\) being isomorphic to \(A_4\), \(S_4\) or \(A_5\) remain for examination.\N\NFor these situations necessary and sufficient conditions are given for \(\mathcal O_K\) to be a free module over \(\mathfrak A _{K/\mathbb Q}\) (Theorems 1.7, 1.8 and 1.9). The results depend on the ramification behaviour of the prime \(2\), and in case of \(G \simeq A_5\) also of the primes \(3\) and \(5\).\N\NTo check local freeness, the proof has to handle several cases. Furthermore, a Magma implementation of an algorithm of \textit{T. Hofmann} and \textit{H. Johnston} [Math. Comput. 89, No. 326, 2931--2963 (2020; Zbl 1452.11134)] is used to check isomorphisms between lattices.