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Zbl 0993.11046
Fouvry, Étienne
Sur les propriétés de divisibilité des nombres de classes des corps quadratiques. (On divisibility properties of class numbers of quadratic fields.). (French)
[J] Bull. Soc. Math. Fr. 127, No.1, 95-113 (1999). ISSN 0037-9484

Let $h(\Delta)$ denote the class number in the usual sense of a quadratic number field with discriminant $\Delta$. The author's main result is the following: there exist infinitely many positive squarefree $\Delta \equiv 1 \bmod 4$ such that $h(\Delta)$ is odd, $\Delta + 4$ is squarefree, and $h(\Delta+4)$ is not divisible by $3$. As the author points out, this result exhibits a `certain independence of the $2$-rank of $\Bbb Q(\sqrt{\Delta})$ and the $3$-rank of $\Bbb Q(\sqrt{\Delta + 4})$'. The ingredients for the quite involved proof come from the theory of character sums, sieves, and cubic forms.
[Franz Lemmermeyer (San Marcos)]

MSC 2000:: *11N35 Sieves
11R29 Class numbers, class groups, discriminants
11R11 Quadratic extensions
11T23 Exponential sums

Keywords: class number of quadratic field; fundamental discriminants; genus theory; sieves; exponential sums

Cited in: Zbl 0986.11054

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