## Proceedings of the Japan Academy, Series A, Mathematical Sciences

### Imaginary cyclic fields of degree $p - 1$ whose relative class numbers are divisible by $p$

Yasuhiro Kishi

#### Abstract

We give a sufficient condition for an imaginary cyclic field of degree $p - 1$ containing $\mathbf{Q}(\zeta + \zeta^{-1})$ to have the relative class number divisible by $p$. As a consequence, we see that there exist infinitely many imaginary cyclic fields of degree $p - 1$ with the relative class number divisible by $p$.

#### Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 77, Number 4 (2001), 55-58.

Dates
First available in Project Euclid: 23 May 2006

https://projecteuclid.org/euclid.pja/1148393081

Digital Object Identifier
doi:10.3792/pjaa.77.55

Mathematical Reviews number (MathSciNet)
MR1829375

Zentralblatt MATH identifier
1006.11063

#### Citation

Kishi, Yasuhiro. Imaginary cyclic fields of degree $p - 1$ whose relative class numbers are divisible by $p$. Proc. Japan Acad. Ser. A Math. Sci. 77 (2001), no. 4, 55--58. doi:10.3792/pjaa.77.55. https://projecteuclid.org/euclid.pja/1148393081

#### References

• Herz, C. S.: Construction of class fields. Seminar on Complex Multiplication: Seminar held at the Institute for Advanced Study, Princeton, N.J., 1957-58. (eds. Borel, A., Chowla, S., Herz, C. S., Iwasawa, K., and Serre, J.-P.). Lecture Notes in Math., no. 21, Springer, Berlin-Heidelberg-New York, pp. VII-1–VII-21 (1966).
• Imaoka, M., and Kishi, Y.: Spiegelung Relations Between Dihedral Extensions and Frobenius Extensions. Tokyo Metropolitan Univ.Math. Preprint Series, no. 12, (2000).
• Katayama, S.: On fundamental units of real quadratic fields with norm $+1$. Proc. Japan Acad., 68A, 18–20 (1992).
• Nagel, Tr.: Über die Klassenzahl imaginär-quadratischer Zahlköper. Abh. Math. Sem. Univ. Hamburg, 1, 140–150 (1922).
• Nakano, S.: On the construction of certain number fields. Tokyo J. Math., 6, 389–395 (1983).
• Parry, C. J.: Real quadratic fields with class numbers divisible by five. Math. Comp., 32, 1261–1270 (1978).
• Sase, M.: On a family of quadratic fields whose class numbers are divisible by five. Proc. Japan Acad., 74A, 120–123 (1998).
• Satgé, M.: Corps résolubles et divisibilité de nombres de classes d'idéaux. Enseign. Math.(2), 25, 165–188 (1979).
• Uehara, T.: On class numbers of cyclic quartic fields. Pacific J. Math., 122, 251–255 (1986).