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Ideal class groups of cyclotomic number fields. II. (English) Zbl 0901.11031
The author continues his study of the ideal class groups of cyclotomic fields and of more general CM-fields [cf. Acta Arith. 72, 347-359 (1995; Zbl 0837.11059)]. By using methods that primarily come from class field theory he proves a lot of new results, many of which strenghten or extend previously known results. In some cases he just gives a new elegant proof for a known theorem.
To quote some of the results, denote by \(\zeta_m\) a primitive \(m\)th root of 1, by \(k^+\) the maximal real subfield of an abelian field \(k\), by Cl\((K)\) the ideal class group of a number field \(K\), and by Cl\(_p(K)\) the \(p\)-part of this group (\(p\) a prime). Among other things, the author proves the following. (i) If \(m\) is divisible by three distinct primes \(\equiv 1 \pmod 4\), then the class number of \(\mathbb{Q}(\zeta_m)^+\) is even. (ii) For \(p>2\), let \(k\) be a complex subfield of \(\mathbb{Q}(\zeta_p)\) and let \(K^+\) be an abelian unramified extension of \(k^+\) with \((K^+:k^+)=n\). Then Cl\(_2(kK^+)\) contains a subgroup of type \((\mathbb{Z}/ 2\mathbb{Z})^{n-1}\). Furthermore, the author finds cyclic quartic subfields of \(\mathbb{Q}(\zeta_p)\) having infinite class field tower. He studies the capitulation of ideal classes in cyclic \(p\)-extensions \(K/k\) of CM-fields, formulating several necessary and sufficient conditions for the minus part of the capitulation kernel for \(K/k\) to be trivial.
In the final section he obtains results about the behaviour of class groups under the natural transfer map \(j: \text{Cl}(k) \to \text{Cl}(K)\) when \(K/k\) is as above. One result states that \(\text{Cl}_p(k)^j = \text{Cl}_p(K)^p\) if \(\text{Cl}_p(k)^j\) and \(\text{Cl}_p(K)\) have the same rank.

MSC:
11R18 Cyclotomic extensions
11R29 Class numbers, class groups, discriminants
11R21 Other number fields
11R37 Class field theory
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