Abstract
The purpose of this article is to determine the monogenity of families of certain biquadratic fields K and cyclic bicubic fields L obtained by composition of the quadratic field of conductor 5 and the simplest cubic fields over the field Q of rational numbers applying cubic Gauß sums. The monogenic biquartic fields K are constructed without using the integral bases. It is found that all the bicubic fields L over the simplest cubic fields are non-monogenic except for the conductors 7 and 9. Each of the proof is obtained by the evaluation of the partial differents !"!# of the different !F/Q(") with F=K or L of a candidate number !, which will or would generate a power integral basis of the fields F.Here ! denotes a suitable Galois action of the abelian extensions F/Q and !F/Q(") is defined by !"G\{#}$(%&%!),where G and ! denote respectively the Galois group of F/Q and the identity embedding of F.
References
Ahmad S, Nakahara T, Hameed A. On certain pure sextic fields related to a problem of Hasse. International Journal of Algebra and Computation 2016; 26-3: 577-563.
Ahmad S, Nakahara T, Husnine SM. Power integral bases for certain pure sextic fields. International Journal of Number Theory (Singapore) 2014; 10(8): 2257-2265. https://doi.org/10.1142/S1793042114500778
Akizuki S, Ota K. On power bases for ring of integers of relative Galois extensions. Bull London Math Soc 2013; 45: 447-452. https://doi.org/10.1112/blms/bds112
Dedekind R. Über die Zusammenhang zwischen der Theorie der Ideals und der Theorie der höhren Kongruenzen. Abh Akad Wiss Göttingen Math-Phys Kl 1878; 23: 1-23.
Gaál I. Diophantine equations and power integral bases, new computational methods, Birkhäuser Boston, Inc., Boston, 2002. https://doi.org/10.1007/978-1-4612-0085-7
Gras M-N, Tanoé F. Corps biquadratiques monogènes. Manuscripta Math 1995; 86: 63-77. https://doi.org/10.1007/BF02567978
Györy K. Discriminant form and index form equations, Algebraic Number Theory and Diophantine Analysis (F. Halter-Koch and R. F. Tichy. Eds.), Walter de Gruyter, Berlin-New York, 2000; 191-214.
Hameed A, Nakahara T. Integral basis and relative monogenity of pure octic fields. Bull Math Soc Sci Math Roumani 2015; 58-4: 419-433.
Hameed A, Nakahara T, Husnine S, Ahmad S. On existing of canonical number system in certain class of pure algebraic number fields Journal of Prime Research in Mathematics 2011; 7: 19-24.
Katayama S-I. On the Class Numbers of Real Quadratic Fields of Richaud-Degest Type. J Math Tokushima Univ 1997; 31: 1-6.
Khan N, Nakahara T, Katayama S-I, Uehara T. Monogenity of totally real algebraic extension fields over a cyclotomic field. Journal of Number Theory 2016; 158: 348-355. https://doi.org/10.1016/j.jnt.2015.06.018
Khan N, Nakahara T. On the cyclic sextic fields of prime conductor related to a problem of Hasse. To be submitted.
Montgomery L, Weinberger P. Real Quadratic Fields with Large Class Number. Math Ann 1977; 225: 173-176. https://doi.org/10.1007/BF01351721
Motoda Y. Notes on quartic fields. Rep Fac Sci Engrg Saga U Math 2003; 32-1: 1-19. Appendix and Crrigenda to "Notes on Quartic Fields," ibid, 37-1 (2008) 1-8.
Motoda Y, Nakahara T. Power integral basis in algebraic number fields whose galois groups are 2-elementry abelian. Arch Math (Basel) 2004; 83: 309-316. https://doi.org/10.1007/s00013-004-1077-0
Motoda Y, Nakahara T, Shah SIA. On a problem of Hasse for certain imaginary abelian fields. J Number Theory 2002; 96: 326-334. https://doi.org/10.1006/jnth.2002.2805
Motoda Y, Nakahara T, Shah SIA, Uehara T. On a problem of Hasse, RIMS kokyuroku Bessatsu. Kyoto Univ B 2009; 12: 209-221.
Nagell T. Zur Arithmetik der Polynome. Abh Math Sem Hamburg 1922; 1: 180-194. https://doi.org/10.1007/BF02940590
Nakahara T. On cyclic biquadratic fields related to a problem of Hasse. Mh Math 1982; 94: 125-132. https://doi.org/10.1007/BF01301930
Narkiewicz W. Elementary and Analytic Theory of Algebraic Numbers, Springer-Verlag, 1st ed. 1974, 3rd ed. Berlin-Heidelberg-New York; PWM-Polish Scientific Publishers, Warszawa 2007.
Ricci G. Ricerche arithmetiche sur polinome. Rend Circ Mat Palermo 1933; 57: 433-475. https://doi.org/10.1007/BF03017586
Shanks D. The simplest cubic fields. Mathematics of Computation 1974; 28-128: 1137-1152.
Sultan M, Nakahara T. On certain octic biquartic fields related to a problem of Hasse. Monatshefte für Mathmatik 2014; 174(4): 153-162.
Sultan M, Nakahara T. Monogenity of biquadratic fields related to Dedekind-Hasse’s problem. Punjab University Journal of Mathematics 2015; 47(2): 77-82.
Washington LC. Introduction to cyclotomic fields, Graduate texts in mathematics, 2nd ed., Springer-Verlag, New York-Heidelberg-Berlin 1995; 83.
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
Copyright (c) 2018 Journal of Basic & Applied Sciences