Minus class groups of the fields of the \(l\)-th roots of unity. (English) Zbl 0902.11043

Schoof, René

Math. Comput. 67, No. 223, 1225-1245 (1998).

Let \(K\) be a CM-field, let \(\iota\) denote complex conjugation, and let \(K^+\) be the fixed field of \(\iota\). There are two different notions of minus class groups in the literature: the group \(\text{Cl}^-(K)\), defined as the kernel of the norm map \(N: \text{Cl}(K) \rightarrow \text{Cl}(K^+)\), and the quotient \(\text{Cl}^*(K)\) of \(\text{Cl}(K)\) by the image of \(\text{Cl}(K^+)\) under the transfer of ideal classes. In the main part of this paper, the author studies the field \(K\) of the \(l\)-th roots of unity for odd primes \(l\); here the two notions coincide.
Let \(G\) denote the Galois group of \(K/\mathbb{Q}\), and let \(\widehat{\mathbb{Z}} = \prod_p \mathbb{Z}_p\) be the Prüfer ring. In Theorem I, the author realizes \(\text{Cl}^*(K)\) as the cokernel of an injection \(\Theta: L \rightarrow L\), where \(L\) is a free \(\widehat{\mathbb{Z}}[G]/(1+\iota)\)-module of finite rank. In Theorem II, he shows that \(L\) may be taken to be of rank \(1\) for all \(l \leq 509\), and that \(\Theta\) can be identified with a modified Stickelberger element. In particular, \(\text{Cl}^*(K)\) is isomorphic to \(\widehat{\mathbb{Z}}[G]/(1+\iota, \Theta)\) as a \(\widehat{\mathbb{Z}}[G]/(1+\iota)\)-module.
The proof of Theorem I combines deep results from Iwasawa theory, notably an extension due to D. R. Solomon [Ann. Inst. Fourier 40, No. 3, 467-492 (1990; Zbl 0694.12004)] of the main conjecture by Mazur and Wiles, plus corresponding results of C. Greither [Ann. Inst. Fourier 42, No. 3, 449-500 (1991; Zbl 0757.11039)] to cover the \(2\)-part, not to mention local and global class field theory. For studying the case \(p = 2\), the author uses his cohomological approach from [J. Pure Appl. Algebra 53, 125-137 (1988; Zbl 0651.12004)]. Here it becomes clear why he prefers \(\text{Cl}^*(K)\) to \(\text{Cl}^-(K)\): it turns out that the cohomology of \(\text{Cl}^*(K)\) behaves much more nicely than that of \(\text{Cl}^-(K)\).
For a proof of Theorem II, one needs the complete factorization of the relative class numbers for primes \(l \leq 509\), and these factorizations are given in an Appendix, including information about the contributions coming from subfields. Other tables give the structure of \(\text{Cl}^*(K)\) as an abelian group for every \(l \leq 509\).
The author’s Theorem III is much more elementary and gives a sufficient criterion for \(\text{Cl}_p^*(K)\) to be cyclic. Related results have been published by K. Tateyama [Proc. Japan Acad., Ser. A 58, 333-335 (1982; Zbl 0509.12005)] and F. Lemmermeyer [Acta Arith. 84, 59-70 (1998)].

Reviewer: Franz Lemmermeyer (Saarbrücken)

MSC:

11R18	Cyclotomic extensions
11R29	Class numbers, class groups, discriminants
11R34	Galois cohomology
11-04	Software, source code, etc. for problems pertaining to number theory
11R23	Iwasawa theory
11B68	Bernoulli and Euler numbers and polynomials

Keywords:

minus class groups; Bernoulli numbers; Stickelberger elements; group cohomology; fields of the \(l\)th roots of unity; Iwasawa theory; factorization of the relative class numbers; tables

Citations:

Zbl 0694.12004; Zbl 0757.11039; Zbl 0651.12004; Zbl 0509.12005

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