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Powerful necessary conditions for class number problems. (English) Zbl 0871.11078
The author intends to determine the CM-fields whose ideal class groups are of exponent at most two, or quartic non-CM-fields of class number one. Lately, two class number problems on imaginary abelian number fields have been solved. One of them is to determine all imaginary abelian number fields with class number one by K. Yamamura [cf. Math. Comput. 62, 899-921 (1994; Zbl 0798.11046)], and another one is to determine all non-quadratic imaginary cyclic number fields of 2-power degree with ideal class groups of exponent at most two by S. Louboutin [cf. Math. Comput. 64, 323-340 (1995; Zbl 0822.11072)]. In both cases, the first step is to get an explicit upper bound on the conductors of the number field under consideration, and the next step is to compute the relative class numbers of all such imaginary abelian number fields with conductor less than this upper bound.
In this paper, the author gives a necessary condition for the ideal class group of a CM-field to have exponent at most two, which enables us to reduce the amount of this relative class number computation. He also gives a necessary condition for some quartic non-CM-fields to have class number one.
Reviewer: H.Yokoi (Iwasaki)

MSC:
11R29 Class numbers, class groups, discriminants
11R21 Other number fields
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