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Exponents of the class groups of imaginary abelian number fields. (English) Zbl 0597.12006
The conjecture is advanced that there exist only finitely many imaginary abelian number fields which have class group of exponent 2. This has been proved by I. Chowla [Q. J. Math., Oxf. Ser. 5, 304-307 (1934; Zbl 0010.33705)] for imaginary quadratic fields and by the author and O. H. Körner [Proc. Am. Math. Soc. 86, 196-198 (1982; Zbl 0502.12005)] for imaginary quadratic extensions of any fixed totally real number field.
The article includes a survey of related results as well as some new results on the conjecture. The main theorem of the article states that for any positive constants \(\alpha\) and \(\delta\) there exist only finitely many imaginary abelian fields K with class group of exponent 2 and with \(h\leq \alpha d^{1/2-\delta}\). Here h and d denote the class number and the discriminant, respectively, of the maximal totally real subfield of K.
The article concludes with results, which are stronger than the conjecture for quartic fields and certain cyclotomic fields.
Reviewer: Ch.Parry

MSC:
11R18 Cyclotomic extensions
11R23 Iwasawa theory
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