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On ideal class groups of algebraic number fields. (English) Zbl 0559.12004

The purpose of this paper is to prove the following theorem (and some related results): given 3 natural numbers \(r_ 1\), \(r_ 2\), and \(n\), consider number fields of degree \(r_ 1+2 r_ 2\) and with \(r_ 1\) real embeddings. In this family there exists an infinite subfamily of fields whose ideal class group contains a subgroup isomorphic to the product of \(r_ 2+1\) copies of \(\mathbb Z/n\mathbb Z\). For example: there exist infinitely many totally real number fields of given degree with their class numbers divisible by \(n\).
The proof consists in considering fields defined by a root of an equation of type \(\prod^{r_ 1+2r_ 2}_{0}(X-A_ i)+C^ n=0\): the condition on the class number and the number of real embeddings is translated in nine different conditions. A step by step method proves that one can verify the conditions for infinitely many fields; one of the main tool is the density theorem about prime numbers.
Reviewer: R.Gillard

MSC:

11R29 Class numbers, class groups, discriminants
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