On an obstruction to the Hasse norm principle and the equality of norm groups of algebraic number fields
\textit{Yu. A. Drakokhrust} and \textit{V. P. Platonov} [Math. USSR, Izv. 29, 299-322 (1987; Zbl 0624.12009)] defined the first obstruction to Hasse norm principle for \(L/K\) to a Galois tower \(k\subset L\subset E\) of algebraic number fields. In this paper the author generalizes this notion to finite extensions \(L/k\) and \(T/k\) of algebraic number fields: The first obstruction to Hasse norm principle for \(L/k\) corresponding to \(T/k\) is defined as the factor group of \(k^*\cap N_{L/k} J_L N_{T/k} J_T\) by \((k^*\cap N_{T/k} J_T) N_{L/k} L^*\), where \(J_L\) and \(J_T\) denote the idele groups of \(L\) and \(T\), respectively. The first section studies this group in general, and proves that this group is computed in finite group theory using decomposition groups of some finitely many primes of \(k\) in a finite Galois extension of \(k\) which contains \(L/k\) and \(T/k\). If for any finite extension \(L/k\) the equality \(N_{K/k} K^*= N_{L/k} L^*\) implies that \(K\) is conjugate to \(L\) over \(k\), then a finite extension \(K\) of \(k\) is called \(k\)-solitary. In a preceding paper [J. Number Theory 50, 1-32 (1995; Zbl 0822.11075)] \textit{R. Guralnick} and the author studied solitary Galois extensions and gave a criterion for an extension to be solitary in the case of Galois extensions of degrees \(\leq 8\). The second section of the present paper applies the results of the first section to solitary non-Galois extensions of degrees \(\leq 5\), and studies when two finite extensions have equal norm groups. In particular, some explicit conditions for such non-Galois extensions to be solitary are obtained.
scientific article; zbMATH DE number 1275045
scientific article