## Nagoya Mathematical Journal

### Monogenesis of the rings of integers in certain imaginary abelian fields

#### Abstract

In this paper we consider a subfield $K$ in a cyclotomic field $k_m$ of conductor $m$ such that $\left[k_m : K\right] = 2$ in the cases of $m = \ell p^n$ with a prime $p,$ where $\ell = 4$ or $p > \ell = 3.$ Then the theme is to know whether the ring of integers in $K$ has a power basis or does not.

#### Article information

Source
Nagoya Math. J., Volume 168 (2002), 85-92.

Dates
First available in Project Euclid: 27 April 2005

https://projecteuclid.org/euclid.nmj/1114631781

Mathematical Reviews number (MathSciNet)
MR1942395

Zentralblatt MATH identifier
1036.11052

Subjects
Primary: 11R18: Cyclotomic extensions
Secondary: 11R04: Algebraic numbers; rings of algebraic integers

#### Citation

Shah, Syed Inayat Ali; Nakahara, Toru. Monogenesis of the rings of integers in certain imaginary abelian fields. Nagoya Math. J. 168 (2002), 85--92. https://projecteuclid.org/euclid.nmj/1114631781

#### References

• Dummit, D. S. and Kisilevsky, H., Indices in cyclic cubic fields, Number Theory and Algebra , Collect. Pap. Dedic. H. B. Mann, A. E. Ross and O. Taussky-Todd, New York San Francisco London, Academic Press, 1977, 29–42.
• Gaál, I., Computing all power integral bases in orders of totally real cyclic sextic number fields , Math. Comp., 65 (1996), 801–822.
• Gras, M.-N., Non monogénéité de l'anneau des extensions cycliques de $\mathbb Q$ de degré premier $\ell \geq 5,$ , J. Number Theory, 23 (1986), 347–353.
• Huard, J. G., Spearman, B. K. and Williams, K. S., Integral Bases for Quartic Fields with Quadratic Subfields , J. Number Theory, 51 (1995), 87–102.
• Liang, J., On integral basis of the maximal real subfield of a cyclotomic field , J. Reine Angew. Math., 286/287 (1976), 223–226.
• Nakahara, T., On cyclic biquadratic fields related to a problem of Hasse , Mh. Math., 94 (1982), 125–132.
• ––––, A simple proof for non-monogenesis of the rings of integers in some cyclic Fields , the Proceedings of the third Conference of the Canadian Number Theory Association, Oxford, Clarendon Press, 1993, 167–173.
• Narkiewicz, W., Elementary and Analytic Theory of Algebraic Numbers, 2nd Edition (1990, Berlin Heidelberg New York, Springer-Verlag; Warszawa, PWN-Polish Scientific Publishers).
• Shah, S. I. A., and Nakahara, T., Non-monogenetic aspect of the rings of integers in certain abelian fields , the Proceedings of the Jangjeon Mathematical Society, Pusan, Ku-Deok Co.,1, 2000, 75–79.
• Thérond, J. -D., Existence d'une extension cyclique monogéne de discriminant donné , Arch. Math., 41 (1983), 243–255.
• Washington, L. C., Introduction to cyclotomic fields, 2nd Edition GTM 83 1997, New York Heidelberg Berlin, Springer-Verlag..