An ideal theoretic proof on monogenity of cyclic sextic fields of prime power conductor. (English) Zbl 1428.11186
It is shown that there exactly three monogenic cyclic sextic fields of prime-power conductor, namely \(\mathbb Q(\zeta_7), \mathbb Q(\zeta_9)\) and the maximal real subfield of \(\mathbb Q(\zeta_{13})\).
Reviewer: Władysław Narkiewicz (Wrocław)
MSC:
| 11R06 | PV-numbers and generalizations; other special algebraic numbers; Mahler measure |
| 11R20 | Other abelian and metabelian extensions |
References:
| [1] | Ahmad, S.; Nakahara, T.; Husnine, S., Power integral bases for certain pure sextic fields, Int. J. Number Theory, 10, 219-226 (2014) |
| [2] | Akizuki, S.; Ota, K., On power bases for rings of integers of relative Galois extensions, Bull. Lond. Math. Soc., 45, 447-452 (2013) · Zbl 1318.11135 |
| [3] | Dedekind, R., Über die Zusammenhang zwischen der Theorie der Ideals und der Theorie der höhren Kongruenzen, Abh.-Akad. Wiss. Goettin., Math. Phys. Kl., 23, 1-23 (1878) |
| [4] | Dummit, D. S.; Kisilevsky, H., Indices in cyclic cubic fields, (Number Theory and Algebra, Collection of Papers Dedicated to H.B. Mann, A.E. Ross and O. Taussky-Todd (1977), Academic Press: Academic Press New York/San Francisco/London), 29-42 · Zbl 0377.12003 |
| [5] | Gaál, I., Diophantine Equations and Power Integral BasesNew Computational Methods (2002), Birkhäuser Boston, Inc.: Birkhäuser Boston, Inc. Boston · Zbl 1016.11059 |
| [6] | Gras, M.-N., Non monogénéité de l’anneau des entier de extensions cycliques de \(Q\) de degreé premier \(\ell \geqq 5\), J. Number Theory, 23, 347-353 (1986) |
| [7] | Gras, M.-N.; Tanoé, F., Corps biquadratiques monogènes, Manuscripta Math., 86, 63-77 (1995) · Zbl 0816.11058 |
| [8] | Hameed, A.; Nakahara, T., Integral bases and relative monogenity of pure octic fields, Bull. Math. Soc. Sci. Math. Roumanie, 58(106), 4, 419-433 (2015) · Zbl 1363.11094 |
| [9] | Hasse, H., Vorlesungen über Zahlentheorie (1964), Springer Verlag: Springer Verlag Berlin, Göttingen, Heidelberg, New York, §20 · Zbl 0123.04201 |
| [10] | Ireland, K.; Rosen, M., A Classical Introduction to Modren Number Theory (1972), Springer-Verlag: Springer-Verlag New York, Berlin, Heidelberg, London, Paris, Tokyo, Hong Kong, 2nd ed. 1990 |
| [11] | N. Khan, On the monogenity of cyclic sextic fields of prime conductor, Thesis, submitted for publication to NUCES, Lahore campus, xiv+82.; N. Khan, On the monogenity of cyclic sextic fields of prime conductor, Thesis, submitted for publication to NUCES, Lahore campus, xiv+82. |
| [12] | Leopoldt, H. W., Über die Hauptordnung der ganzen Elemente eienes abelishen Zahlkörpers, J. Reine Angew., 201, 119-149 (1959) · Zbl 0098.03403 |
| [13] | Motoda, Y., Notes on quartic fields, Rep. Fac. Sci. Eng. Saga Univ., Math.. Rep. Fac. Sci. Eng. Saga Univ., Math., Rep. Fac. Sci. Eng. Saga Univ., Math., 37, 1, 1-8 (2008), Appendix and corrigenda to “Notes on Quartic Fields” · Zbl 1158.11344 |
| [14] | Motoda, Y.; Nakahara, T.; Shah, S. I.A.; Uehara, T., On a problem of Hasse, RIMS Kôkyûroku Bessatsu, 12, 209-221 (2009) · Zbl 1251.11073 |
| [15] | Nakahara, T., On cyclic biquadratic fields related to a problem of Hasse, Monatsh. Math., 94, 125-132 (1982) · Zbl 0482.12001 |
| [16] | Nakahara, T., On the indices and integral bases of non-cyclic but abelian biquadratic fields, Arch. Math. (Basel), 41, 504-508 (1983) · Zbl 0513.12005 |
| [17] | Nakahara, T., A simple proof of non-monogenesis of the ring of integers in some cyclic fields, (Advances in Number Theory (1993), Kingston Claredon press), 167-173 · Zbl 0797.11089 |
| [18] | Narkiewicz, W., Elementary and Analytic Theory of Algebraic Numbers (2007), Springer-Verlag/PWM-Polish Scientific Publishers: Springer-Verlag/PWM-Polish Scientific Publishers Berlin, Heidelberg, New York/Warszawa |
| [19] | Shah, S. I.A., Monogenesis of the ring of integers in a cyclic sextic field of prime conductor, Rep. Fac. Sci. Eng. Saga Univ., Math., 29, 1-10 (2000) · Zbl 0952.11026 |
| [20] | Shah, S. I.A.; Nakahara, T., Monogenesis of the rings of integers in certain imaginary abelian fields, Nagoya Math. J., 168, 85-92 (2002) · Zbl 1036.11052 |
| [21] | Shimura, G., Arithmetic of Quadratic Forms, Springer Monograph in Mathematics (2010), Springer · Zbl 1202.11026 |
| [22] | Yamamura, K., Bibliography on Monogenity of Orders of Algebraic Number Fields (November 2017), National Defense Academy of Japan, updated ed. available on demand [171 papers with MR#/Zbl# are included] |
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