Kurihara, Masato On the ideal class groups of the maximal real subfields of number fields with all roots of unity. (English) Zbl 0949.11055 J. Eur. Math. Soc. (JEMS) 1, No. 1, 35-49 (1999). Let \(k\) be a totally real number field and \(k_\infty= \bigcup_{n>0} k(\mu_n)\) be the field obtained by adjoining all the roots of unity to \(k\), and \(k_\infty^+\) the maximal real subfield of \(k_\infty\). In this paper, the author shows the nice main result that the ideal class group \(C_{k_\infty^+}\) of \(k_\infty^+\) is trivial. The point of the proof is that \(k_\infty\) contains all \(\mu_l\) for any prime \(l\). By choosing a “special” prime \(l\) (infinitely many such primes exist), the author succeeds in making a specific ideal class capitulate in a totally real subfield of the field obtained by adjoining \(\mu_l\). On the other hand, as the author mentions, we know that the ideal class group \(C_{k_\infty}\) is generated by infinitely many elements [e.g. A. Brumer, J. Pure Appl. Algebra 20, 107-111 (1981; Zbl 0476.12006), K. Horie, Compos. Math. 74, 1-14 (1990; Zbl 0701.11063), or the preprint of M. Kurihara referred to as [11] in the paper under review]. This result reminds us of Greenberg’s conjecture, which states that the ideal class group of the cyclotomic \(\mathbb{Z}_p\)-extension of \(k\) is trivial [R. Greenberg, Am. J. Math. 98, 263-284 (1976; Zbl 0334.12013)]. Actually, the author applies his method to the conjecture, and proves some results on the existence of infinitely many real abelian extensions of degree divisible by \(p^n\) for which Greenberg’s conjecture for \(p\) is true, where \(p\) is a fixed odd prime and \(n\) is any natural number. Further, in an Appendix, the author shows as a CM-analogue of the main theorem that \(C_{Kk^{ab}}\) is trivial if \(k\) is an imaginary quadratic field and \(k^{ab}\) is the maximal abelian extension of \(k\), and \(K\) is any number field. As an immediate consequence of it, the author obtains a conjecture of Gras [Conjecture 0.5 in G. Gras, J. Number Theory 62, 403-421 (1997; Zbl 0869.11088)] is valid for number fields including an imaginary quadratic field. Reviewer: Hisao Taya (Sendai) Cited in 4 ReviewsCited in 13 Documents MSC: 11R23 Iwasawa theory 11R29 Class numbers, class groups, discriminants 11R80 Totally real fields 11R34 Galois cohomology Keywords:Iwasawa theory; class groups; maximal real subfields; Galois cohomology; existence of infinitely many real abelian extensions; Greenberg’s conjecture Citations:Zbl 0476.12006; Zbl 0701.11063; Zbl 0334.12013; Zbl 0869.11088 PDFBibTeX XMLCite \textit{M. Kurihara}, J. Eur. Math. Soc. (JEMS) 1, No. 1, 35--49 (1999; Zbl 0949.11055) Full Text: DOI