## 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

### Keywords:

class numbers; abelian extensions

### Citations:

Zbl 0788.11052; Zbl 0914.11057
Full Text: