On the class number and unit index of simplest quartic fields. (English) Zbl 0719.11073

The simplest quartic fields are defined by adjoining to \({\mathbb{Q}}\) a root of \(X^ 4-tX^ 3-6X^ 2+tX+1\), \(t\in {\mathbb{Z}}^+\), where \(t^ 2+16\) is not divisible by an odd square. This article contains a complete list of simplest quartic fields of even conductor and class number at most two. For even conductor (corresponding to even t), the class number is one if and only if \(t\in \{2,4,6,8,10,24\}\). The class number is two if and only if \(t\in \{12,16,20\}\). The class number of the quadratic subfield is one in all these cases except \(t=16\), where it is two. The author determines the Hasse unit index Q for simplest fields except the case where the fundamental unit of the quadratic subfield has norm -1 and the conductor is an odd composite.
Reviewer: A.J.Lazarus


11R29 Class numbers, class groups, discriminants
11R27 Units and factorization
11R16 Cubic and quartic extensions
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