A family of pairs of imaginary cyclic fields of degree $(p-1)/2$ with both class numbers divisible by $p$

Miho Aoki; Yasuhiro Kishi

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arXiv · 1809.07982

A family of pairs of imaginary cyclic fields of degree $(p-1)/2$ with both class numbers divisible by $p$

Miho Aoki·Yasuhiro Kishi

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Abstract

We construct a new infinite family of pairs of imaginary cyclic fields of degree $(p-1)/2$ explicitly with both class numbers divisible by a given prime number $p$. For the proof, we use the fundamental unit of $\mathbb Q(\sqrt{p})$, certain units which are roots of a parametric quartic polynomial, the Kummer theory, the Gauss sums and the Jacobi sums, linear recurrence sequences, a consequence of the Weil conjecture and a result of Lenstra which is a generalization of Artin conjecture on primitive roots. Our result is based on the famous Scholz' results on pairs of quadratic fields $\mathbb Q(\sqrt{D})$ and $\mathbb Q (\sqrt{-3D})$.

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Miho Aoki, Yasuhiro Kishi. 2018-09-21. A family of pairs of imaginary cyclic fields of degree $(p-1)/2$ with both class numbers divisible by $p$. https://arxiv.org/abs/1809.07982

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