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On \(p\)-class group of an \(A_n\)-extension. (English) Zbl 1192.11077

Let \(p\) be a prime and \(L\) an \(A_n\)-extension over a number field \(K\). The aim of this paper is to estimate the ratio of the \(p\)-class number of \(L\) to the ambiguous \(p\)-class number of \(L\) with respect to \(K\).
Theorem. Let \(L\) be a finite Galois extension over \(K\) an algebraic number field of finite degree. Assume \(n\geq 5\) and \(\text{Gal}(L/K)\) is isomorphic to \(A_n\), the alternating group of degree \(n\). Let \(\ell\) be the maximal prime number satisfying \(\ell\neq p\) and \(\ell\leq\sqrt n\). If \(h_L\{p\} > a_{L/K}\) then \(h_L\{p\}/a_{L/K}\) is divisible by \(p^{\ell+1}\).
This implies a theorem of K. Ohta [J. Math. Soc. Japan 30, No. 4, 763–770 (1978; Zbl 0389.12002)].

MSC:

11R29 Class numbers, class groups, discriminants
11R32 Galois theory
11R21 Other number fields

Citations:

Zbl 0389.12002
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Full Text: DOI Euclid

References:

[1] K. Ohta, On the \(p\)-group of a Galois number field and its subfields, J. Math. Soc. Japan 30 (1978), no. 4, 763-770. · Zbl 0389.12002
[2] G. Cornell and M. Rosen, Group-theoretic constrains on the structure of the class groups, J. Number Theory 13 (1981), no. 1, 1-11. · Zbl 0456.12005
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