×

Integral bases and monogenity of composite fields. (English) Zbl 1490.11106

Summary: We consider infinite parametric families of high degree number fields composed of quadratic fields with pure cubic, pure quartic, pure sextic fields and with the so called simplest cubic, simplest quartic fields. We explicitly describe an integral basis of the composite fields. We construct the index form, describe their factors and prove that the monogenity of the composite fields imply certain divisibility conditions on the parameters involved. These conditions usually cannot hold, which implies the non-monogenity of the fields. The fields that we consider are higher degree number fields, of degrees 6 up to 12. The non-monogenity of the number fields is stated very often as a consequence of the non-existence of the solutions of the index form equation. As per our knowledge, it is not at all feasible to solve the index form equation in these high degree fields, especially not in a parametric form. On the other hand, our method implies directly the non-monogenity in almost all cases. We obtain our results in a parametric form, characterizing these infinite parametric families of composite fields.

MSC:

11R04 Algebraic numbers; rings of algebraic integers
11R16 Cubic and quartic extensions
11R21 Other number fields
11Y50 Computer solution of Diophantine equations

Software:

Maple
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Chang, [Chang 02] M.-L., Non-Monogenity in a Family of Sextic Fields,, J. Number Theory., 97, 252-268 (2002)
[2] Char, [Char 88] B. W.; Geddes, K. O.; Gonnet, G. H.; Monagan, M. B.; Watt (Eds.), S. M., MAPLE, Reference Manual (1988), Watcom Publications: Watcom Publications, Waterloo, Canada
[3] Cohen, [Cohen 93] H., A Course in Computational Algebraic Number Theory, Graduate Texts in Mathematics, 138 (1993), Berlin: Springer, Berlin
[4] Cook, [Cook] J. P., Computing Integral Bases
[5] Gaál, [Gaál 98] I., Power Integral bases in Composites of Number Fields, Canad. Math. Bulletin, 41, 158-165 (1998) · Zbl 0951.11012
[6] Gaál, [Gaál 02] I., Diophantine Equations and Power Integral Bases (2002), Boston: Birkhäuser, Boston · Zbl 1016.11059
[7] Gaál, [Gaál And Olajos 03] I.; Olajos, P., Recent Results on Power Integral bases of Composite Fields, Acta Acad. Paedagog. Agriensis, Sect. Mat. (N.S.), 30, 45-54 (2003) · Zbl 1101.11012
[8] Gaál, [Gaál Et Al. 02] I.; Olajos, P.; Pohst, M., Power Integer bases in Orders of Composite Fields, Experimental Math., 11, 87-90 (2002) · Zbl 1020.11064
[9] Gaál, [Gaál And Remete 17] I.; Remete, L., Integral bases and Monogenity of Pure Fields, J. Number Theory, 173, 129-146 (2017) · Zbl 1419.11118
[10] Gaál, [Gaál And Remete] I.; Remete, L., Non-Monogenity in a Family of Octic Fields, 47, 817-824 (2017) · Zbl 1381.11102
[11] Gaál, [Gaál And Remete] I.; Remete, L., Integral Bases and Monogenity of the Simplest Sextic Fields · Zbl 1409.11085
[12] Gras, [Gras 77-78] M. N., Table Numerique du nombre de classes et des unites des extensions cycliques reelles de degré 4 in Q, Publ. Math. Fac. Sci. Besancon · Zbl 0471.12006
[13] Kim, [Kim And Kim 03] H. K.; Kim, J. S., Computation of the Different of the Simplest Quartic Fields, Manuscript (2003)
[14] Narkiewicz, [Narkiewicz 90] W., Elementary and Analytic Theory of Algebraic Numbers (1990), Springer · Zbl 0717.11045
[15] Olajos, [Olajos 03] P., Power Integral bases in Orders of Composite Fields. II, Ann. Univ. Sci. Budap. Rolando Etvs, Sect. Math., 46, 35-41 (2003) · Zbl 1077.11023
[16] Shanks, [Shanks 74] D., The Simplest Cubic Fields, Math. Comput., 28, 1137-1152 (1974) · Zbl 0307.12005
[17] Stewart, [Stewart And Tall 16] I.; Tall, D., Algebraic Number Theory and Fermat’s Last Theorem (2016), Boca Raton, FL: CRC Press, Boca Raton, FL · Zbl 1332.11001
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.