A family of pairs of imaginary cyclic fields of degree $$(p-1)/2$$(p-1)/2 with both class numbers divisible by p
| Autor: | Miho Aoki, Yasuhiro Kishi |
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| Rok vydání: | 2019 |
| Předmět: |
Class (set theory)
Algebra and Number Theory Degree (graph theory) Mathematics::Number Theory 010102 general mathematics Prime number 0102 computer and information sciences 01 natural sciences Combinatorics Integer 010201 computation theory & mathematics 0101 mathematics Mathematics Characteristic polynomial Fundamental unit (number theory) |
| Zdroj: | The Ramanujan Journal. 52:133-161 |
| ISSN: | 1572-9303 1382-4090 |
| DOI: | 10.1007/s11139-018-0085-9 |
| Popis: | Let p be a prime number with $$p\equiv 5\ (\mathrm{mod}\ {8})$$. We construct a new infinite family of pairs of imaginary cyclic fields of degree $$(p-1)/2$$ with both class numbers divisible by p. Let $$k_0$$ be the unique subfield of $$\mathbb {Q}(\zeta _p)$$ of degree $$(p-1)/4$$ and $$u_p=(t+b\sqrt{p})/2\,(>1)$$ be the fundamental unit of $$k:=\mathbb {Q}(\sqrt{p})$$. We put $$D_{m,n}:={\mathcal {L}}_m(2{\mathcal {F}}_m-{\mathcal {F}}_n{\mathcal {L}}_m)b$$ for integers m and n, where $$\{ {\mathcal {F}}_n \}$$ and $$\{ {\mathcal {L}}_n \}$$ are linear recurrence sequences of degree two associated to the characteristic polynomial $$P(X)=X^2-tX-1$$. We assume that there exists a pair $$(m_0,n_0)$$ of integers satisfying certain congruence relations. Then we show that there exists a positive integer $$N_q$$ which satisfies the both class numbers of $$k_0(\sqrt{D_{m,n}})$$ and $$k_0(\sqrt{pD_{m,n}})$$ are divisible by p for any pairs (m, n) with $$m\equiv m_0 \ (\mathrm{mod}\ {N_q}), \ n\equiv n_0 \ (\mathrm{mod}\ {N_q})$$ and $$n>3$$. Furthermore, we show that if we assume that ERH holds, then there exists the pair $$(m_0,n_0)$$. |
| Databáze: | OpenAIRE |
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