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On the class number of some real abelian number fields of prime conductors. (English) Zbl 1210.11116

For an odd prime \(p\), let \(K\) be a real abelian field of conductor \(p\) and of prime degree \(l=(p-1)/4\). The author proves that the class number of \(K\) is divisible by no prime \(q\) which is a primitive root modulo \(l\) and belongs to the range \(3<q<\sqrt{p}\). The same result holds for \(l=(p-1)/6\) when \(q\) is restricted to the range \(3<q<\sqrt{p}/2\). The approach is based on the author’s earlier articles [Abh. Math. Semin. Univ. Hamb. 63, 67–86 (1993; Zbl 0788.11052)] and [Math. Comput. 67, No. 221, 369–398 (1998; Zbl 0914.11057)]. If Schinzel’s conjecture about linear polynomials holds true then for each prime \(q\) there are infinitely many primes \(p\) satisfying the assumptions in these results.

MSC:

11R29 Class numbers, class groups, discriminants
11R20 Other abelian and metabelian extensions
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