## 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

Zbl 0389.12002
Full Text:

### 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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