Solitary Galois extensions of algebraic number fields
Let \(k\) be an algebraic number field. For any finite Galois extensions \(K/k\) and \(L/k\), \textit{L. Stern} [J. Number Theory 32, 203-219 (1989; Zbl 0687.12006)] has proved that the norm groups \(N_{K/k} K^ \times\) and \(N_{L/k} L^ \times\) are equal iff \(K=L\). A finite extension \(K\) of \(k\) is called ``\(k\)-solitary'' if for any finite extension \(L\) of \(k\) the equality \(N_{K/k} K^ \times= N_{L/k} L^ \times\) implies that \(K\) and \(L\) are conjugate over \(k\) (i.e. \(k\)-isomorphic). The classification problem of \(k\)-solitary extensions seems to be difficult to solve in general. In the present paper, the authors describe finite Galois extensions \(K\) of \(k\) which are \(k\)-solitary by first proving that \(K/k\) is necessarily a 2-extension. They state that \(k\) itself and every quadratic extension \(K/k\) are \(k\)-solitary. They provide a necessary condition for a finite Galois 2-extension \(K/k\) with \([K: k]\geq 4\) to be \(k\)-solitary. Afterwards, they investigate covering subgroups and show that this necessary condition is also sufficient for extensions of degree 4 or 8. In the last section, a dihedral Galois extension \(E/\mathbb{Q}\) of degree 8, with \(\mathbb{Q}\)-solitary field \(E\), is constructed.
scientific article; zbMATH DE number 714107