Hasse’s problem for monogenic fields
Annales Mathématiques Blaise Pascal, Tome 16 (2009) no. 1, pp. 47-56.

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].

DOI : https://doi.org/10.5802/ambp.252
Classification : 11R27,  11R29,  11R37
Mots clés : remplir svp
@article{AMBP_2009__16_1_47_0,
     author = {Nakahara, Toru},
     title = {Hasse's problem for monogenic fields},
     journal = {Annales Math\'ematiques Blaise Pascal},
     pages = {47--56},
     publisher = {Annales math\'ematiques Blaise Pascal},
     volume = {16},
     number = {1},
     year = {2009},
     doi = {10.5802/ambp.252},
     mrnumber = {2514526},
     zbl = {1187.11038},
     language = {en},
     url = {www.numdam.org/item/AMBP_2009__16_1_47_0/}
}
Nakahara, Toru. Hasse’s problem for monogenic fields. Annales Mathématiques Blaise Pascal, Tome 16 (2009) no. 1, pp. 47-56. doi : 10.5802/ambp.252. http://www.numdam.org/item/AMBP_2009__16_1_47_0/

[1] Dummit, D. S.; Kisilevsky, H. Indices in cyclic cubic fields, Number theory and algebra, Academic Press, New York, 1977, pp. 29-42 | MR 460272 | Zbl 0377.12003

[2] Gaál, I. Diophantine equations and power integral bases, Birkhäuser Boston Inc., Boston, MA, 2002 (New computational methods) | MR 1896601 | Zbl 1016.11059

[3] Gras, M.-N. Non monogénéité de l’anneau des entiers des extensions cycliques de Q de degré premier l5, J. Number Theory, Volume 23 (1986) no. 3, pp. 347-353 | Article | MR 846964 | Zbl 0582.12003

[4] Gras, M.-N.; Tanoé, F. Corps biquadratiques monogènes, Manuscripta Math., Volume 86 (1995) no. 1, pp. 63-79 | Article | MR 1314149 | Zbl 0816.11058

[5] Motoda, Y. Notes on Quartic Fields, Rep. Fac. Sci. Engrg. Saga Univ. Math., Volume 32-1 (2003), pp. 1-19 (Appendix and corrigenda to “Notes on Quartic Fields”, ibid., 37-1(2008), 1–8.) | MR 2017249

[6] Motoda, Y.; Nakahara, T. 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 (2004), pp. 91-99 | MR 2109026

[7] Motoda, Y.; Nakahara, T. Power integral bases in algebraic number fields whose Galois groups are 2-elementary abelian, Arch. Math., Volume 83 (2004), pp. 309-316 | Article | MR 2096803 | Zbl 1078.11061

[8] Motoda, Y.; Nakahara, T.; Shah, S.I.A. On a problem of Hasse for certain imaginary abelian fields, J. Number Theory, Volume 96 (2002), pp. 326-334 ([cf. http://dlwww.dl.saga-u.ac.jp/contents/diss/GI00000879/motodaphd.pdf ]) | MR 1932459 | Zbl 1032.11043

[9] Motoda, Y.; Park, K.H.; Nakahara, T. On power integral bases of the 2-elementary abelian extension fields, Trends in Mathematics, Volume 9-1 (2006), pp. 55-63

[10] Nakahara, T. On Cyclic Biquadratic Fields Related to a Problem of Hasse, Mh. Math., Volume 94 (1982), pp. 125-132 | Article | MR 678047 | Zbl 0482.12001

[11] Nakahara, T. On the Indices and Integral Bases of Non-cyclic but Abelian Biquadratic Fields, Arch. Math., Volume 41 (1983), pp. 504-508 | Article | MR 731633 | Zbl 0513.12005

[12] Nakahara, T. On the Indices and Integral Bases of Abelian Biquadratic Fields, RIMS Kōkyūroku, Distribution of values of arithmetic functions, Volume 517 (1984), pp. 91-100

[13] Nakahara, T. On the Minimum Index of a Cyclic Quartic Field, Arch. Math., Volume 48 (1987), pp. 322-325 | Article | MR 884563 | Zbl 0627.12003

[14] Nakahara, T. 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, New York, 1993, pp. 167-173 | MR 1368417 | Zbl 0797.11089

[15] Park, K.H.; Motoda, Y.; Nakahara, T. On integral bases of certain octic abelian fields (Submitted)

[16] Shah, S.I.A. Monogenesis of the ring of integers in a cyclic sextic field of a prime conductor, Rep. Fac. Sci. Engrg. Saga Univ. Math., Volume 29-1 (2000), pp. 1-10 | MR 1769574 | Zbl 0952.11026

[17] Shah, S.I.A.; Nakahara, T. Monogenesis of the rings of integers in certain imaginary abelian fields, Nagoya Math. J., Volume 168 (2002), pp. 85-92 | MR 1942395 | Zbl 1036.11052