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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
##### Keywords:
Galois extension fields
##### Citations:
Zbl 0139.28104; Zbl 0428.12003
Full Text:
##### 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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