Hasse’s problem for monogenic fields

Toru Nakahara[1]

  • [1] Department of Mathematics, Faculty of Science and Engineering, Saga University, Saga 840-8502, Japan. Current address: NUCES, Peshawar Campus, 160-Industrial Estate, Hayatabad, Peshawar, N.W.F.P. The Islamic Republic of Pakistan

Annales mathématiques Blaise Pascal (2009)

  • Volume: 16, Issue: 1, page 47-56
  • ISSN: 1259-1734

Abstract

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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].

How to cite

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Nakahara, Toru. "Hasse’s problem for monogenic fields." Annales mathématiques Blaise Pascal 16.1 (2009): 47-56. <http://eudml.org/doc/10571>.

@article{Nakahara2009,
abstract = {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].},
affiliation = {Department of Mathematics, Faculty of Science and Engineering, Saga University, Saga 840-8502, Japan. Current address: NUCES, Peshawar Campus, 160-Industrial Estate, Hayatabad, Peshawar, N.W.F.P. The Islamic Republic of Pakistan},
author = {Nakahara, Toru},
journal = {Annales mathématiques Blaise Pascal},
keywords = {Power integral basis; monogenic fields; Hasse’s problem; power integral basis},
language = {eng},
month = {1},
number = {1},
pages = {47-56},
publisher = {Annales mathématiques Blaise Pascal},
title = {Hasse’s problem for monogenic fields},
url = {http://eudml.org/doc/10571},
volume = {16},
year = {2009},
}

TY - JOUR
AU - Nakahara, Toru
TI - Hasse’s problem for monogenic fields
JO - Annales mathématiques Blaise Pascal
DA - 2009/1//
PB - Annales mathématiques Blaise Pascal
VL - 16
IS - 1
SP - 47
EP - 56
AB - 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].
LA - eng
KW - Power integral basis; monogenic fields; Hasse’s problem; power integral basis
UR - http://eudml.org/doc/10571
ER -

References

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  1. D. S. Dummit, H. Kisilevsky, Indices in cyclic cubic fields, Number theory and algebra (1977), 29-42, Academic Press, New York Zbl0377.12003MR460272
  2. I. Gaál, Diophantine equations and power integral bases, (2002), Birkhäuser Boston Inc., Boston, MA Zbl1016.11059MR1896601
  3. M.-N. Gras, Non monogénéité de l’anneau des entiers des extensions cycliques de Q de degré premier l ≥ 5 , J. Number Theory 23 (1986), 347-353 MR846964
  4. M.-N. Gras, F. Tanoé, Corps biquadratiques monogènes, Manuscripta Math. 86 (1995), 63-79 Zbl0816.11058MR1314149
  5. Y. Motoda, Notes on Quartic Fields, Rep. Fac. Sci. Engrg. Saga Univ. Math. 32-1 (2003), 1-19 MR2017249
  6. Y. Motoda, 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 (2004), 91-99 Zbl1078.11061MR2109026
  7. Y. Motoda, T. Nakahara, Power integral bases in algebraic number fields whose Galois groups are 2 -elementary abelian, Arch. Math. 83 (2004), 309-316 Zbl1078.11061MR2096803
  8. Y. Motoda, T. Nakahara, S.I.A. Shah, On a problem of Hasse for certain imaginary abelian fields, J. Number Theory 96 (2002), 326-334 Zbl1032.11043MR1932459
  9. Y. Motoda, K.H. Park, T. Nakahara, On power integral bases of the 2 -elementary abelian extension fields, Trends in Mathematics 9-1 (2006), 55-63 
  10. T. Nakahara, On Cyclic Biquadratic Fields Related to a Problem of Hasse, Mh. Math. 94 (1982), 125-132 Zbl0482.12001MR678047
  11. T. Nakahara, On the Indices and Integral Bases of Non-cyclic but Abelian Biquadratic Fields, Arch. Math. 41 (1983), 504-508 Zbl0513.12005MR731633
  12. T. Nakahara, On the Indices and Integral Bases of Abelian Biquadratic Fields, RIMS Kōkyūroku, Distribution of values of arithmetic functions 517 (1984), 91-100 
  13. T. Nakahara, On the Minimum Index of a Cyclic Quartic Field, Arch. Math. 48 (1987), 322-325 Zbl0627.12003MR884563
  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) (1993), 167-173, Oxford Univ. Press, New York Zbl0797.11089MR1368417
  15. K.H. Park, Y. Motoda, T. Nakahara, On integral bases of certain octic abelian fields Zbl1158.11342
  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 (2000), 1-10 Zbl0952.11026MR1769574
  17. S.I.A. Shah, T. Nakahara, Monogenesis of the rings of integers in certain imaginary abelian fields, Nagoya Math. J. 168 (2002), 85-92 Zbl1036.11052MR1942395

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