Note on the rings of integers of certain tame 2-Galois extensions over a number field
\noindent A finite group \(G\) and a number field \(k\) are said to satisfy the condition \(A(G,k)\) if every tame Galois extension \(K/k\) with group \(G\) has an integral normal basis. Theorem: \(A(D_4,k)\Leftrightarrow A(\mathbb{Z}/2\times\mathbb{Z}/2,k)\) (with \(D_4\) the dihedral group of order 8). The theorem generalizes \textit{H. Ichimura} [Proc. Japan Acad., Ser. A 78, No. 6, 76--79 (2002; Zbl 1106.11308)], by which \([A(\mathbb{Z}/2\times\mathbb{Z}/2,k)\Leftrightarrow h_4(k)=1]\) where \(h_4(k)\) denotes the order of the ray class group of \(k\) defined modulo \(4\mathfrak{o}_k\), and also \textit{N. P. Byott} and \textit{B. Sodaïgui} [J. Number Theory 112, No. 1, 1--19 (2005; Zbl 1073.11068)], by which \([h_4(k)=1\Rightarrow A(D_4,k)]\). Related work is, first of all, \textit{L. R. McCulloh} [J. Algebra 82, 102--134 (1983; Zbl 0508.12008)], and moreover, \textit{H. Ichimura} and \textit{H. Sumida-Takahashi} [J. Math. Soc. Japan 58, No. 3, 885--902 (2006; Zbl 1102.11059)], \textit{C. Greither} et al. [J. Number Theory 79, No. 1, 164--173 (1999; Zbl 0941.11044)]. The paper finishes with an application concerning the \(n\)-th layer field \(k_n\) of the cyclotomic \(\mathbb{Z}_2\)-extension of \(\mathbb{Q}\). Then \(A(\mathbb{Z}/2\times\mathbb{Z}/2,k_n)\) and \(A(D_4,k_n)\) hold if the class number \(h_n^+\) of \(k_n\) is trivial; moreover, every tame 2-Abelian extension of \(k_n\) of exponent 2 has an integral normal basis.
scientific article; zbMATH DE number 5184528