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)].


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


Zbl 0389.12002
Full Text: DOI Euclid


[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.