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Nakahara, Toru
Hasse’s problem for monogenic fields. Annales mathématiques Blaise Pascal, 16 no. 1 (2009), p. 47-56
Full text djvu | pdf | Reviews MR 2514526 | Zbl 1187.11038
Class. Math.: 11R27, 11R29, 11R37

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In this article we shall give a survey of Hasse’s problem for integral power bases of algebraic number fields during the last half of century. Specifically, we developed this problem for the abelian number fields and we shall show several substantial examples for our main theorem [7] [9], which will indicate the actual method to generalize for the forthcoming theme on Hasse’s problem [15].


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[6] Y. Motoda and T. Nakahara, Monogenesis of Algebraic Number Fields whose Galois Groups are $2$-elementary Abelian, Proceedings of the 2003 Nagoya Conference “Yokoi-Chowla Conjecture and Related Problems”, Edited by S.-I. Katayama, C. Levesque and T. Nakahara, Furukawa Total Pr.Co. Saga, 91-99, 2004  MR 2109026
[7] Y. Motoda and T. Nakahara, Power integral bases in algebraic number fields whose Galois groups are $2$-elementary abelian, Arch. Math., 83:309-316, 2004  MR 2096803 |  Zbl 1078.11061
[8] Y. Motoda, T. Nakahara and S.I.A. Shah, On a problem of Hasse for certain imaginary abelian fields, J. Number Theory, 96:326-334, 2002  MR 1932459 |  Zbl 1032.11043
[9] Y. Motoda, K.H. Park and T. Nakahara, On power integral bases of the $2$-elementary abelian extension fields, Trends in Mathematics, 9-1:55-63, 2006
[10] T. Nakahara, On Cyclic Biquadratic Fields Related to a Problem of Hasse, Mh. Math., 94:125-132, 1982  MR 678047 |  Zbl 0482.12001
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[13] T. Nakahara, On the Minimum Index of a Cyclic Quartic Field, Arch. Math., 48:322-325, 1987  MR 884563 |  Zbl 0627.12003
[14] T. Nakahara, A simple proof for non-monogenesis of the rings of integers in some cyclic fields, Advances in number theory (Kingston, ON, 1991), Oxford Sci. Publ., Oxford Univ. Press, 1993  MR 1368417 |  Zbl 0797.11089
[15] K.H. Park, Y. Motoda and T. Nakahara, On integral bases of certain octic abelian fields, Submitted
[16] S.I.A. Shah, Monogenesis of the ring of integers in a cyclic sextic field of a prime conductor, Rep. Fac. Sci. Engrg. Saga Univ. Math., 29-1:1-10, 2000  MR 1769574 |  Zbl 0952.11026
[17] S.I.A. Shah and T. Nakahara, Monogenesis of the rings of integers in certain imaginary abelian fields, Nagoya Math. J., 168:85-92, 2002
Article |  MR 1942395 |  Zbl 1036.11052
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