On the ideal class groups of the maximal real subfields of number fields with all roots of unity
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. \textit{A. Brumer}, J. Pure Appl. Algebra 20, 107-111 (1981; Zbl 0476.12006), \textit{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 [\textit{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 \textit{G. Gras}, J. Number Theory 62, 403-421 (1997; Zbl 0869.11088)] is valid for number fields including an imaginary quadratic field.
existence of infinitely many real abelian extensions
scientific article; zbMATH DE number 1264484