Imaginary cyclic fields of degree \(p-1\) whose relative class numbers are divisible by \(p\). (English) Zbl 1006.11063

Kishi, Yasuhiro

Proc. Japan Acad., Ser. A 77, No. 4, 55-58 (2001).

Let \(p\) be an odd prime, \(\zeta\) be a primitive \(p\)th root of unity and \(\omega= \zeta+ \zeta^{-1}\). Let \(k= \mathbb{Q} (\sqrt{d})\) be a real quadratic field that is not contained in \(\mathbb{Q} (\zeta)\). Then \(\mathbb{Q} (\zeta, \sqrt{d})\) is a bicyclic biquadratic extension of \(\mathbb{Q}(\omega)\) and so contains a unique subfield \(M\) different from \(\mathbb{Q}(\zeta)\) and \(k(\omega)\) that is a quadratic extension of \(\mathbb{Q}(\omega)\). The main result of the article is that if there exists a unit \(\varepsilon\) of \(k\) with \(\varepsilon^p \not\in k^p\) which satisfies \(\text{Tr} (\varepsilon)\equiv 0\pmod {p^2}\) then the relative class number of \(M\) is divisible by \(p\). The cases \(p=3\) and \(p=5\) are included in the work of C. S. Herz [Lect. Notes Math. 21, VII-1–VII-21 (1966; Zbl 0147.03902)] and the reviewer [Math. Comput. 32, 1261-1270 (1978; Zbl 0401.12008)].
The author uses a result of M. Sase [Proc. Japan Acad., Ser. A 74, 120-123 (1998; Zbl 0926.11081)] to explicitly construct an Abelian unramified extension \(E\) of \(M\) when the hypothesis of the theorem is satisfied. It is also shown that the Galois group of \(E/Q\) is the Frobenius group \(F_p\) of order \(p(p-1)\). When \(p\equiv 3\pmod 4\), \(p\) divides the class number of \(k\) and the hypothesis of the theorem is satisfied, the previously mentioned fact enables the author to show that the \(p\)-class group has rank at least 2. For \(p=7\), the author gives a table of all \(d\leq 5000\) for which the class number of \(k\) is divisible by 7 and the hypothesis of the theorem is satisfied. There are 28 such fields and for each one the structure of the class group of \(M\) is given.

Reviewer: Charles Parry (Blacksburg)

MSC:

11R29	Class numbers, class groups, discriminants
11R20	Other abelian and metabelian extensions
12F10	Separable extensions, Galois theory

Keywords:

cyclic field; Frobenius group; relative class number

Citations:

Zbl 0147.03902; Zbl 0401.12008; Zbl 0926.11081

References:

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