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Keywords:
discrete valuation ring; Dedekind ring; monogenity; relative integral basis; nonic field
Summary:
Let $L = K(\alpha )$ be an extension of a number field $K$, where $\alpha $ satisfies the monic irreducible polynomial $P(X)=X^{p}-\beta $ of prime degree belonging to $\mathfrak {o}_{K}[X]$ ($\mathfrak {o}_K$ is the ring of integers of $K$). The purpose of this paper is to study the monogenity of $L$ over $K$ by a simple and practical version of Dedekind's criterion characterizing the existence of power integral bases over an arbitrary Dedekind ring by using the Gauss valuation and the index ideal. As an illustration, we determine an integral basis of a pure nonic field $L$ with a pure cubic subfield, which is not necessarily a composite extension of two cubic subfields. We obtain a slightly simpler computation of the discriminant $d_{L/\mathbb {Q}}$.
References:
[1] Atiyah, M. F., Macdonald, I. G.: Introduction to Commutative Algebra. Addison-Wesley, Massachusetts (1969). DOI 10.1201/9780429493621 | MR 0242802 | Zbl 0175.03601
[2] Cassels, J. W. S., (eds.), A. Fröhlich: Algebraic Number Theory. Academic Press, London (1967). MR 0215665 | Zbl 0153.07403
[3] Cassou-Noguès, P., Taylor, M. J.: A note on elliptic curves and the monogeneity of rings of integers. J. Lond. Math. Soc., II. Ser. 37 (1988), 63-72. DOI 10.1112/jlms/s2-37.121.63 | MR 0921747 | Zbl 0639.12001
[4] Cassou-Noguès, P., Taylor, M. J.: Unités modulaires et monogénéité d'anneaux d'entiers. Séminaire de théorie des nombres, Paris 1986-87 Progress in Mathematics 75. Birkhäuser, Boston (1988), 35-64 French. MR 0990505 | Zbl 0714.11078
[5] Charkani, M. E., Deajim, A.: Generating a power basis over a Dedekind ring. J. Number Theory 132 (2012), 2267-2276. DOI 10.1016/j.jnt.2012.04.006 | MR 2944754 | Zbl 1293.11101
[6] Charkani, M. E., Deajim, A.: Relative index extensions of Dedekind rings. JP J. Algebra Number Theory Appl. 27 (2012), 73-84. MR 3086201 | Zbl 1368.11111
[7] Charkani, M. E., Lahlou, O.: On Dedekind's criterion and monogenicity over Dedekind rings. Int. J. Math. Math. Sci. 2003 (2003), 4455-4464. DOI 10.1155/S0161171203211534 | MR 2040142 | Zbl 1066.11046
[8] Charkani, M. E., Sahmoudi, M.: Sextic extension with cubic subfield. JP J. Algebra Number Theory Appl. 34 (2014), 139-150. Zbl 1307.11112
[9] Charkani, M. E., Sahmoudi, M., Soullami, A.: Tower index formula and monogenecity. Commun. Algebra 49 (2021), 2469-2475. DOI 10.1080/00927872.2021.1872590 | MR 4255019 | Zbl 1470.11268
[10] Cohen, H.: A Course in Computational Algebraic Number Theory. Graduate Texts in Mathematics 138. Springer, Berlin (1993). DOI 10.1007/978-3-662-02945-9 | MR 1228206 | Zbl 0786.11071
[11] Dedekind, R.: Über den Zusammenhang zwischen der Theorie der Ideale und der Theorie der höheren Congruenzen. Abh. Akad. Wiss. Gött., Math-Phys. Kl., 3. Folge 23 (1878), 3-38 German.
[12] Fröhlich, A., Taylor, M. J.: Algebraic Number Theory. Cambridge Studies in Advanced Mathematics 27. Cambridge University Press, Cambridge (1993). DOI 10.1017/CBO9781139172165 | MR 1215934 | Zbl 0744.11001
[13] Gaál, I., Remete, L.: Power integral bases in cubic and quartic extensions of real quadratic fields. Acta Sci. Math. 85 (2019), 413-429. DOI 10.14232/actasm-018-080-z | MR 4154697 | Zbl 1449.11104
[14] Hameed, A., Nakahara, T.: Integral bases and relative monogenity of pure octic fields. Bull. Math. Soc. Sci. Math. Roum., Nouv. Sér. 58 (2015), 419-433. MR 3443598 | Zbl 1363.11094
[15] Ichimura, H.: On power integral bases of unramified cyclic extensions of prime degree. J. Algebra 235 (2001), 104-112. DOI 10.1006/jabr.2000.8467 | MR 1807657 | Zbl 0972.11101
[16] Janusz, G. J.: Algebraic Number Fields. Graduate Studies in Mathematics 7. AMS, Providence (1996). DOI 10.1090/gsm/007 | MR 1362545 | Zbl 0854.11001
[17] Kumar, M., Khanduja, S. K.: A generalization of Dedekind criterion. Commun. Algebra 35 (2007), 1479-1486. DOI 10.1080/00927870601168897 | MR 2317622 | Zbl 1145.11078
[18] Lavallee, M. J., Spearman, B. K., Williams, K. S.: Lifting monogenic cubic fields to monogenic sextic fields. Kodai Math. J. 34 (2011), 410-425. DOI 10.2996/kmj/1320935550 | MR 2855831 | Zbl 1237.11045
[19] Mann, H. B.: On integral bases. Proc. Am. Math. Soc. 9 (1958), 167-172. DOI 10.1090/S0002-9939-1958-0093502-7 | MR 0093502 | Zbl 0081.26602
[20] Narkiewicz, W.: Elementary and Analytic Theory of Algebraic Numbers. Springer, Berlin (1990). DOI 10.1007/978-3-662-07001-7 | MR 1055830 | Zbl 0717.11045
[21] Neukirch, J.: Algebraic Number Theory. Grundlehren der Mathematischen Wissenschaften 322. Springer, Berlin (1999). DOI 10.1007/978-3-662-03983-0 | MR 1697859 | Zbl 0956.11021
[22] Sahmoudi, M.: Explicit integral basis for a family of sextic field. Gulf J. Math. 4 (2016), 217-222. DOI 10.56947/gjom.v4i4.280 | MR 3603475 | Zbl 1366.11107
[23] Sahmoudi, M., Soullami, A.: On monogenicity of relative cubic-power extensions. Adv. Math., Sci. J. 9 (2020), 6817-6827. DOI 10.37418/amsj.9.9.40
[24] Sahmoudi, M., Soullami, A.: On sextic integral bases using relative quadratic extention. Bol. Soc. Parana. Mat. (3) 38 (2020), 175-180. DOI 10.5269/bspm.v38i4.40042 | MR 3912302 | Zbl 1431.35011
[25] Schmid, P.: On criteria by Dedekind and Ore for integral ring extensions. Arch. Math. 84 (2005), 304-310. DOI 10.1007/s00013-004-1227-4 | MR 2135040 | Zbl 1072.13004
[26] Soullami, A., Sahmoudi, M., Boughaleb, O.: On relative power integral basis of a family of numbers fields. Rocky Mt. J. Math. 51 (2021), 1443-1452. DOI 10.1216/rmj.2021.51.1443 | MR 4298858 | Zbl 1469.11414
[27] Spearman, B. K., Williams, K. S.: Relative integral bases for quartic fields over quadratic subfields. Acta Math. Hung. 70 (1996), 185-192. DOI 10.1007/BF02188204 | MR 1374384 | Zbl 30864.11051
[28] Spearman, B. K., Williams, K. S.: A relative integral basis over $\mathbb{Q}(\sqrt{-3})$ for the normal closure of a pure cubic field. Int. J. Math. Math. Sci. 25 (2003), 1623-1626. DOI 10.1155/s0161171203204336 | MR 1979698 | Zbl 1064.11070
[29] Washington, L. C.: Relative integral bases. Proc. Am. Math. Soc. 56 (1976), 93-94. DOI 10.1090/S0002-9939-1976-0399041-9 | MR 0399041 | Zbl 0331.12002
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