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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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