×

On monogenity of certain number fields defined by trinomials. (English) Zbl 1515.11108

A monic irreducible polynomial \(f(x)\) belonging to \(\mathbb{Z}[x]\) having a root \(\theta\) in \(\mathbb{C}\) is said to be monogenic if \(\mathbb{Z}[\theta]\) is the ring of algebraic integers of the algebraic number field \(\mathbb{Q}(\theta)\). An algebraic number field \(K\) with ring of algebraic integers denoted by \(A_{K}\) is said to be monogenic if there exists \(\alpha \in A_{K}\) such that \(A_{K}=\mathbb{Z}[\alpha]\). In this paper, the authors mainly deal with the problem of constructing families of non-monogenic fields which are generated over \(\mathbb{Q}\) by some root of an irreducible trinomial of the type \(f(x)=x^n+ax+b\) belongs to \(\mathbb{Z}[x]\). Examples are also given when a trinomial \(f(x)\) having a root \(\theta\) in \(\mathbb{C}\) is not monogenic but the field \(\mathbb{Q}(\theta)\) is monogenic. The authors use theorem of index of Ore.

MSC:

11R04 Algebraic numbers; rings of algebraic integers
11R21 Other number fields
PDFBibTeX XMLCite
Full Text: DOI arXiv Link

References:

[1] S. Ahmad, T, Nakahara and S.M. Husnine, Power integral bases for certain pure sextic fields, I. J. Number Theory 10(8) (2014), 2257-2265. · Zbl 1316.11094
[2] H. Ben Yakkou, A. Chillali and L. El Fadil, On Power integral bases for certain pure number fields defined by \(x^{2^r \cdot 5^s}- m \), Comm. in Algebra 49(7) (2021), 2916-2926. · Zbl 1471.11260
[3] H. Ben Yakkou and L. El Fadil, On monogenity of certain pure number fields defined by \(x^{p^r}-m\), I. J. Number Theory 7(10) (2021), 2235-2242. · Zbl 1483.11236
[4] H. Ben Yakkou and O. Kchit, On Power integral bases for certain pure number fields defined by \(x^{3^r}- m \), SPJM (2021), DOI: 10.1007/s40863-021-00251-2. · Zbl 1512.11076
[5] Y. Bilu, I. Gaál and K. Györy, Index form equations in sextic fields: a hard computation, Acta Arithmetica 115(1) (2004), 85-96. · Zbl 1064.11084
[6] I.F. Blake, S. Gao and R.C. Mullin, Explicit Factorization of \(x^{2^k}+1\) over \(\mathbb{F}_p\) with prime \(p \equiv 3 \mod 4\), AAECC (4) (1993), 89-94. · Zbl 0778.11069
[7] H. Cohen, A Course in Computational Algebraic Number Theory, GTM 138, Springer-Verlag Berlin Heidelberg, 1993. · Zbl 0786.11071
[8] R R. Dedekind, Über den Zusammenhang zwischen der Theorie der Ideale und der Theorie der höheren Kongruenzen, Göttingen Abhandlungen 23 (1878), 1-23.
[9] L. El Fadil, On integral bases and monogeneity of pure sextic number fields with non-squarefree coefficients, Journal of Number Theory 228 (2021), 375-389. · Zbl 1479.11182
[10] L. El Fadil, On Power integral bases for certain pure sextic fields, Bol. Soc. Paran. Math. 40 (2020), 1-7.
[11] L. El Fadil, On Newton polygon’s techniques and factorization of polynomial over henselian valued fields, J. Algebra Appl. 19(10) (2020), 2050188, DOI: 10.1142/S0219498820501881. · Zbl 1446.13004
[12] L. El Fadil, J. Montes and E. Nart, Newton polygons and \(p\)-integral bases of quartic number fields, J. Algebra Appl. 11(4) (2012), 1250073. · Zbl 1297.11134
[13] A.J. Engler and A. Prestel, Valued fields, Springer-Verlag Berln Heidelberg, 2005. · Zbl 1128.12009
[14] I. Gaál, Diophantine equations and power integral bases, Theory and algorithm, Second edition, Boston, Birkhäuser, 2019. · Zbl 1465.11090
[15] I. Gaál, An experiment on the monogenity of a family of trinomials, JP Journal of Algebra Number Theory Appl. 51(1) (2021), 97-111. · Zbl 1499.11318
[16] I. Gaál and K. Györy, Index form equations in quintic fields, Acta Arithmetica 89(4) (1999), 379-396. · Zbl 0930.11091
[17] I. Gaál, A. Pethö, and M. Pohst, On the resolution of index form equations in quartic number fields, J. Symbolic Comput 16 (1993), 563-584. · Zbl 0808.11023
[18] I. Gaál and L. Remete, Non-monogenity in a family of octic fields, Rocky Mountain J. Math. 47(3) (2017), 817-824. · Zbl 1381.11102
[19] I. Gaál and L. Remete, Power integral bases and monogenity of pure fields, J. Number Theory 173 (2017), 129-146. · Zbl 1419.11118
[20] I. Gaál and N. Schulte, Computing all power integral bases of cubic number fields, Math. Comput. 53 (1989), 689-696. · Zbl 0677.10013
[21] T.A. Gassert, A note on the monogeneity of power maps, Albanian J. Math 11 (2017), 3-12. · Zbl 1392.11082
[22] M.N. Gras, Non monogénéité de l’anneau des entiers des extensions cycliques de \(\mathbb{Q}\) de degré premier \(l \ge 5 \), J. Number Theory 23(3) (1986), 347-353. · Zbl 0564.12008
[23] G.R. Greenfield and D. Drucker, On the discriminant of a trinomial, Linear Algebra and its Applications (62) (1984), 105-112. · Zbl 0552.12013
[24] J. Guàrdia, J. Montes, and E. Nart, Newton polygons of higher order in algebraic number theory, Transactions of the American Mathematical Society 364(1) (2012), 361-416. · Zbl 1252.11091
[25] A. Hameed and T. Nakahara, Integral bases and relative monogenity of pure octic fields, Bull. Math. Soc. Sci. Math. R épub. Soc. Roum 58(106) (2015), 419-433. · Zbl 1363.11094
[26] H. Hasse, Zahlentheorie, Akademie-Verlag, Berlin, 1963. · Zbl 1038.11500
[27] K. Hensel, Theorie der algebraischen Zahlen, Teubner Verlag, Leipzig, Berlin, 1908. · JFM 39.0269.01
[28] K. Hensel, Arithemetishe untersuchungen über die gemeinsamen ausserwesentliche Discriminanten Theiler einer Gattung, J. Reine Angew. Math. 113 (1894), 128-160. · JFM 25.0136.01
[29] K. Hensel, Untersuchung der Fundamentalgleichung einer Gattung für eine reelle Primzahl als Modul und Bestimmung der Theiler ihrer Discriminante, 113 (1894), 61-83. · JFM 25.0135.03
[30] K. Hensel, Arithmetische Untersuchungen über Discriminanten und ihre ausserwesentlichen Theiler, Dissertation, Univ. Berlin, 1884. · JFM 16.0063.01
[31] R. Ibarra, H. Lembeck, M. Ozaslan, H. Smith and K.E. Stange, Monogenic fields arising from trinomials, arXiv:1908.09793v2. · Zbl 1515.11105
[32] B. Jhorar and S.K. Khanduja, On power basis of a class of algebraic number fields, I. J. Number Theory 12(8) (2016), 2317-2321. · Zbl 1357.11106
[33] L. Jones and D. White, Monogenic trinomials with non-squarefree discriminant, International Journal of Mathematics 32(13) (2021), 2150089, 21 pp., DOI: 10.1142/S0129167X21500890. · Zbl 1478.11125
[34] J. Montes and E. Nart, On a theorem of Ore, Journal of Algebra 146(2) (1992), 318-334. · Zbl 0762.11045
[35] Y. Motoda, T. Nakahara and S.I.A. Shah, On a problem of Hasse, J. Number Theory 96 (2002), 326-334. · Zbl 1032.11043
[36] W. Narkiewicz, Elementary and Analytic Theory of Algebraic Numbers, Third Edition, Springer, 2004. · Zbl 1159.11039
[37] A. Pethö and M. Pohst, On the Indices of Multiquadratic Number Fields, Acta Arithmetica 153(4) (2012), 393-414. · Zbl 1255.11052
[38] H. Smith, The monogenity of radical extension, Acta Arithmitica 198 (2021), 313-327. · Zbl 1469.11413
[39] O. Ore, Newtonsche Polygone in der Theorie der algebraischen Korper, Math. Ann. 99 (1928), 84-117. · JFM 54.0191.02
[40] E. Zylinski, Zur Theorie der ausserwesentlichen discriminantenteiler algebraischer Körper, Math. Ann. (73) (1913), 273-274. · JFM 44.0241.02
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.