Gaál, István; Remete, László Integral bases and monogenity of the simplest sextic fields. (English) Zbl 1409.11085 Acta Arith. 183, No. 2, 173-183 (2018). Let \(m\neq -8,-3,0,5\) be a rational integer, assume that \(q_m=m^2+3m+9\) is square-free and denote by \(a_m\) a root of \(x^3-mx^2-(m+3)x-1\). The authors determine an integral basis of the totally real cyclic sextic field \(K_m=Q(\sqrt{q_m},a_m)\) and show that its form depends on \(m\) mod \(36\). They show also the the only monogenic fields \(K_m\) are \(K_{-4}=K_1\) and \(K_{-2}=K_{-1}\) of discriminants \(13^5\) and \(2\cdot14^5\), respectively. Reviewer: Władysław Narkiewicz (Wrocław) Cited in 3 Documents MSC: 11R04 Algebraic numbers; rings of algebraic integers 11R20 Other abelian and metabelian extensions 11Y50 Computer solution of Diophantine equations Keywords:sextic fields; cyclic fields; integral basis; monogenity; power integral basis Software:Maple PDFBibTeX XMLCite \textit{I. Gaál} and \textit{L. Remete}, Acta Arith. 183, No. 2, 173--183 (2018; Zbl 1409.11085) Full Text: DOI arXiv References: [1] B. W. Char, K. O. Geddes, G. H. Gonnet, M. B. Monagan and S. M. Watt (eds.), MAPLE, Reference Manual, Watcom Publ., Waterloo, ON, 1988. · Zbl 0758.68038 [2] H. Cohen, A Course in Computational Algebraic Number Theory, Grad. Texts in Math. 138, Springer, Berlin, 1993. [3] I. Ga´al, Computing all power integral bases in orders of totally real cyclic sextic number fields, Math. Comput. 65 (1996), 801-822. · Zbl 0857.11069 [4] I. Ga´al, Diophantine Equations and Power Integral Bases, Birkh¨auser, Boston, 2002. [5] I. Ga´al, P. Olajos and M. Pohst, Power integer bases in orders of composite fields, Experimental Math. 11 (2002), 87-90. · Zbl 1020.11064 [6] I. Ga´al and G. Petr´anyi, Calculating all elements of minimal index in the infinite parametric family of simplest quartic fields, Czechoslovak Math. J. 64 (139) (2014), 465-475. · Zbl 1340.11102 [7] I. Ga´al and L. Remete, Integral bases and monogenity of pure fields, J. Number Theory 173 (2017), 129-146. · Zbl 1419.11118 [8] M.-N. Gras, Familles d’unit´es dans les extensions cycliques r´eelles de degr´e 6 de Q, in: Publ. Math. Fac. Sci. Besan¸con, Th´eorie des Nombres 1984/85-1985/86, no. 2, 1986, Exp. No. 2, 27 pp. · Zbl 0617.12005 [9] A. Hoshi, On the simplest sextic fields and related Thue equations, Funct. Approx. Comment. Math. 47 (2012), 35-49. · Zbl 1320.11029 [10] G. Lettl, A. Peth˝o and P. Voutier, On the arithmetic of simplest sextic fields and related Thue equations in: Number Theory. Diophantine, Computational and Algebraic Aspects (Eger, 1996), de Gruyter, Berlin, 1998, 331-348. · Zbl 0923.11053 [11] G. Lettl, A. Peth˝o and P. Voutier, Simple families of Thue inequalities, Trans. Amer. Math. Soc. 351 (1999), 1871-1894. · Zbl 0920.11041 [12] W. Narkiewicz, Elementary and Analytic Theory of Algebraic Numbers, 2nd ed., Springer, 1990. · Zbl 0717.11045 [13] M. Pohst and H. Zassenhaus, Algorithmic Algebraic Number Theory, Encyclopedia Math. Appl. 30, Cambridge Univ. Press, 1989. · Zbl 0685.12001 [14] D. Shanks, The simplest cubic fields, Math. Comput. 28 (1974), 1137-1152. · Zbl 0307.12005 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.