# Article

Full entry | PDF   (0.2 MB)
Keywords:
simplest quartic field; power integral base; monogeneity
Summary:
It is a classical problem in algebraic number theory to decide if a number field is monogeneous, that is if it admits power integral bases. It is especially interesting to consider this question in an infinite parametric family of number fields. In this paper we consider the infinite parametric family of simplest quartic fields $K$ generated by a root $\xi$ of the polynomial $P_t(x)=x^4-tx^3-6x^2+tx+1$, assuming that $t>0$, $t\neq 3$ and $t^2+16$ has no odd square factors. In addition to generators of power integral bases we also calculate the minimal index and all elements of minimal index in all fields in this family.
References:
[1] Char, B. W., Geddes, K. O., Gonnet, G. H., Leong, B. L., Monagan, M. B., Watt, S. M.: Maple V---language reference manual. Springer, New York (1991). Zbl 0758.68038
[2] Daberkow, M., Fieker, C., Klüners, J., Pohst, M., Roegner, K., Schörnig, M., Wildanger, K.: KANT V4. J. Symb. Comput. 24 (1997), 267-283. DOI 10.1006/jsco.1996.0126 | MR 1484479 | Zbl 0886.11070
[3] Gaál, I.: Diophantine Equations and Power Integral Bases. New Computational Methods. Birkhäuser, Boston (2002). MR 1896601 | Zbl 1016.11059
[4] Gaál, I., Lettl, G.: A parametric family of quintic Thue equations. II. Monatsh. Math. 131 (2000), 29-35. DOI 10.1007/s006050070022 | MR 1796800 | Zbl 0995.11024
[5] Gaál, I., Pethő, A., Pohst, M.: Simultaneous representation of integers by a pair of ternary quadratic forms---with an application to index form equations in quartic number fields. J. Number Theory 57 (1996), 90-104. DOI 10.1006/jnth.1996.0035 | MR 1378574 | Zbl 0853.11023
[6] Gaál, I., Pohst, M.: Power integral bases in a parametric family of totally real cyclic quintics. Math. Comput. 66 (1997), 1689-1696. DOI 10.1090/S0025-5718-97-00868-5 | MR 1423074 | Zbl 0899.11064
[7] Gras, M. N.: Table numérique du nombre de classes et des unités des extensions cycliques réelles de degré 4 de $\mathbb Q$. French Publ. Math. Fac. Sci. Besançon, Théor. Nombres, Année 1977-1978, Fasc. 2 (1978). MR 0898667
[8] Jadrijević, B.: Establishing the minimal index in a parametric family of bicyclic biquadratic fields. Period. Math. Hung. 58 (2009), 155-180. DOI 10.1007/s10998-009-10155-3 | MR 2531162 | Zbl 1265.11061
[9] Jadrijević, B.: Solving index form equations in the two parametric families of biquadratic fields. Math. Commun. 14 (2009), 341-363. MR 2743182
[10] Kim, H. K., Lee, J. H.: Evaluation of the Dedekind zeta function at $s =-1$ of the simplest quartic fields. Trends in Math., New Ser., Inf. Center for Math. Sci., 11 (2009), 63-79.
[11] Lazarus, A. J.: On the class number and unit index of simplest quartic fields. Nagoya Math. J. 121 (1991), 1-13. MR 1096465 | Zbl 0719.11073
[12] Lettl, G., A.Pethő,: Complete solution of a family of quartic Thue equations. Abh. Math. Semin. Univ. Hamb. 65 (1995), 365-383. DOI 10.1007/BF02953340 | MR 1359142
[13] Lettl, G., Pethő, A., Voutier, P.: Simple families of Thue inequalities. Trans. Am. Math. Soc. 351 (1999), 1871-1894. DOI 10.1090/S0002-9947-99-02244-8 | MR 1487624 | Zbl 0920.11041
[14] Mordell, L. J.: Diophantine Equations. Pure and Applied Mathematics 30 Academic Press, London (1969). MR 0249355 | Zbl 0188.34503
[15] Olajos, P.: Power integral bases in the family of simplest quartic fields. Exp. Math. 14 (2005), 129-132. DOI 10.1080/10586458.2005.10128916 | MR 2169516 | Zbl 1092.11042
[16] Shanks, D.: The simplest cubic fields. Math. Comput. 28 (1974), 1137-1152. DOI 10.1090/S0025-5718-1974-0352049-8 | MR 0352049 | Zbl 0307.12005

Partner of