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On the Galois module structure of ideal class groups. (English) Zbl 1045.11079
Let \(K/k\) be a Galois extension over a number field of degree \(n\) and \(p\) be a prime number which does not divide \(n\). Then, the \(p\)-rank of the ideal class group of \(K\) has been investigated by K. Iwasawa [Nagoya Math. J. 27, 239–247 (1966; Zbl 0139.28104)], J. M. Masley [Compos. Math. 37, 297–319 (1978; Zbl 0428.12003)], and many others.
In this paper, the authors intend to give more general results by removing some conditions assumed there. Namely, denote by \(c(n,p)\) the order of \(p\) in the group \((\mathbb{Z}/n\mathbb{Z})^\times\), and by \(d(n,p)\) the minimum of \(c(\ell,p)\) for all prime factors \(\ell\) of \(n\). Denote by \(C(L)\) and \(h(L)\) the ideal class group and the class number of an algebraic number field \(L\), and for a finite abelian group \(A\), denote by \(r_pA\) its \(p\)-rank. Then, they obtain the following 3 theorems:
Theorem 1: \(r_pC(K)-r_pC(k)\geq d(n,p)\).
Theorem 2: For a cyclic extension \(K/k\) of prime degree \(\ell\neq p\), \(r_pC(K)\equiv r_pC(k)\pmod {c(\ell,p)}\) holds.
Theorem 3. For a solvable extension \(K/k\) of degree \(n\) prime to \(p\), \(r_pC(K)\equiv r_pC(k)\pmod {e(\ell,p)}\) holds, where \(e(\ell,p)\) is the greatest common divisor of \(c(\ell,p)\) for all prime factors \(\ell\) of \(n\).

MSC:
11R29 Class numbers, class groups, discriminants
11R21 Other number fields
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References:
[1] (1992)
[2] Com-positio Math. 37 pp 297– (1978)
[3] DOI: 10.1090/S0025-5718-1988-0929551-0
[4] DOI: 10.1007/BF01399512 · Zbl 0403.12007
[5] (1997)
[6] DOI: 10.1080/00927879508825297 · Zbl 0822.12001
[7] DOI: 10.1016/0022-314X(81)90026-3 · Zbl 0456.12005
[8] NATO adv. Sci. Inst. Ser. C 265 pp 347– (1989)
[9] Advanced topics in computational number theory (2000) · Zbl 0977.11056
[10] Nagoya Math. J. 27 pp 239– (1966) · Zbl 0139.28104
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