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.


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