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Characterization of irreducible algebraic integers by their norms. (English) Zbl 0651.12002

A finite extension L of an algebraic number field K is said to have property \((N^*)\), if for two algebraic integers \(\alpha\), \(\beta\) in L with \(N\alpha\) associated with \(N\beta\) in K, \(\alpha\) and \(\beta\) are either both irreducible or both not. The main result of the paper under review is the following:
“A normal extension \(L| K\) with Galois group G has property \((N^*)\) if and only if one of the following holds: \((a)\quad The\) class group of L written additively is isomorphic to \({\mathbb{Z}}/2{\mathbb{Z}}\oplus {\mathbb{Z}}/2{\mathbb{Z}}\); \((b)\quad G\quad acts\) trivially on the class group of L; \((c)\quad The\) class number of L is odd and there exists an algebraic number field \(L_ 0\), \(K\subseteq L_ 0\subseteq L\) such that \([L_ 0:K]=2\). The author also shows that the property \((N^*)\) depends on the G-module structure of the class group of L written additively.
Reviewer: S.Lal

MSC:

11R04 Algebraic numbers; rings of algebraic integers
11R27 Units and factorization
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