On relative pure cyclic fields with power integral bases

Mohammed Sahmoudi; Mohammed Elhassani Charkani

Mathematica Bohemica (2023)

  • Volume: 148, Issue: 1, page 117-128
  • ISSN: 0862-7959

Abstract

top
Let L = K ( α ) be an extension of a number field K , where α satisfies the monic irreducible polynomial P ( X ) = X p - β of prime degree belonging to 𝔬 K [ X ] ( 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 / ℚ .

How to cite

top

Sahmoudi, Mohammed, and Charkani, Mohammed Elhassani. "On relative pure cyclic fields with power integral bases." Mathematica Bohemica 148.1 (2023): 117-128. <http://eudml.org/doc/299578>.

@article{Sahmoudi2023,
abstract = {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 $d_\{L/\mathbb \{Q\}\}$.},
author = {Sahmoudi, Mohammed, Charkani, Mohammed Elhassani},
journal = {Mathematica Bohemica},
keywords = {discrete valuation ring; Dedekind ring; monogenity; relative integral basis; nonic field},
language = {eng},
number = {1},
pages = {117-128},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On relative pure cyclic fields with power integral bases},
url = {http://eudml.org/doc/299578},
volume = {148},
year = {2023},
}

TY - JOUR
AU - Sahmoudi, Mohammed
AU - Charkani, Mohammed Elhassani
TI - On relative pure cyclic fields with power integral bases
JO - Mathematica Bohemica
PY - 2023
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 148
IS - 1
SP - 117
EP - 128
AB - 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 $d_{L/\mathbb {Q}}$.
LA - eng
KW - discrete valuation ring; Dedekind ring; monogenity; relative integral basis; nonic field
UR - http://eudml.org/doc/299578
ER -

References

top
  1. Atiyah, M. F., Macdonald, I. G., 10.1201/9780429493621, Addison-Wesley, Massachusetts (1969). (1969) Zbl0175.03601MR0242802DOI10.1201/9780429493621
  2. Cassels, J. W. S., (eds.), A. Fröhlich, Algebraic Number Theory, Academic Press, London (1967). (1967) Zbl0153.07403MR0215665
  3. Cassou-Noguès, P., Taylor, M. J., 10.1112/jlms/s2-37.121.63, J. Lond. Math. Soc., II. Ser. 37 (1988), 63-72. (1988) Zbl0639.12001MR0921747DOI10.1112/jlms/s2-37.121.63
  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. (1988) Zbl0714.11078MR0990505
  5. Charkani, M. E., Deajim, A., 10.1016/j.jnt.2012.04.006, J. Number Theory 132 (2012), 2267-2276. (2012) Zbl1293.11101MR2944754DOI10.1016/j.jnt.2012.04.006
  6. Charkani, M. E., Deajim, A., Relative index extensions of Dedekind rings, JP J. Algebra Number Theory Appl. 27 (2012), 73-84. (2012) Zbl1368.11111MR3086201
  7. Charkani, M. E., Lahlou, O., 10.1155/S0161171203211534, Int. J. Math. Math. Sci. 2003 (2003), 4455-4464. (2003) Zbl1066.11046MR2040142DOI10.1155/S0161171203211534
  8. Charkani, M. E., Sahmoudi, M., Sextic extension with cubic subfield, JP J. Algebra Number Theory Appl. 34 (2014), 139-150. (2014) Zbl1307.11112
  9. Charkani, M. E., Sahmoudi, M., Soullami, A., 10.1080/00927872.2021.1872590, Commun. Algebra 49 (2021), 2469-2475. (2021) Zbl1470.11268MR4255019DOI10.1080/00927872.2021.1872590
  10. Cohen, H., 10.1007/978-3-662-02945-9, Graduate Texts in Mathematics 138. Springer, Berlin (1993). (1993) Zbl0786.11071MR1228206DOI10.1007/978-3-662-02945-9
  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. (1878) 
  12. Fröhlich, A., Taylor, M. J., 10.1017/CBO9781139172165, Cambridge Studies in Advanced Mathematics 27. Cambridge University Press, Cambridge (1993). (1993) Zbl0744.11001MR1215934DOI10.1017/CBO9781139172165
  13. Gaál, I., Remete, L., 10.14232/actasm-018-080-z, Acta Sci. Math. 85 (2019), 413-429. (2019) Zbl1449.11104MR4154697DOI10.14232/actasm-018-080-z
  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. (2015) Zbl1363.11094MR3443598
  15. Ichimura, H., 10.1006/jabr.2000.8467, J. Algebra 235 (2001), 104-112. (2001) Zbl0972.11101MR1807657DOI10.1006/jabr.2000.8467
  16. Janusz, G. J., 10.1090/gsm/007, Graduate Studies in Mathematics 7. AMS, Providence (1996). (1996) Zbl0854.11001MR1362545DOI10.1090/gsm/007
  17. Kumar, M., Khanduja, S. K., 10.1080/00927870601168897, Commun. Algebra 35 (2007), 1479-1486. (2007) Zbl1145.11078MR2317622DOI10.1080/00927870601168897
  18. Lavallee, M. J., Spearman, B. K., Williams, K. S., 10.2996/kmj/1320935550, Kodai Math. J. 34 (2011), 410-425. (2011) Zbl1237.11045MR2855831DOI10.2996/kmj/1320935550
  19. Mann, H. B., 10.1090/S0002-9939-1958-0093502-7, Proc. Am. Math. Soc. 9 (1958), 167-172. (1958) Zbl0081.26602MR0093502DOI10.1090/S0002-9939-1958-0093502-7
  20. Narkiewicz, W., 10.1007/978-3-662-07001-7, Springer, Berlin (1990). (1990) Zbl0717.11045MR1055830DOI10.1007/978-3-662-07001-7
  21. Neukirch, J., 10.1007/978-3-662-03983-0, Grundlehren der Mathematischen Wissenschaften 322. Springer, Berlin (1999). (1999) Zbl0956.11021MR1697859DOI10.1007/978-3-662-03983-0
  22. Sahmoudi, M., 10.56947/gjom.v4i4.280, Gulf J. Math. 4 (2016), 217-222. (2016) Zbl1366.11107MR3603475DOI10.56947/gjom.v4i4.280
  23. Sahmoudi, M., Soullami, A., 10.37418/amsj.9.9.40, Adv. Math., Sci. J. 9 (2020), 6817-6827. (2020) DOI10.37418/amsj.9.9.40
  24. Sahmoudi, M., Soullami, A., 10.5269/bspm.v38i4.40042, Bol. Soc. Parana. Mat. (3) 38 (2020), 175-180. (2020) Zbl1431.35011MR3912302DOI10.5269/bspm.v38i4.40042
  25. Schmid, P., 10.1007/s00013-004-1227-4, Arch. Math. 84 (2005), 304-310. (2005) Zbl1072.13004MR2135040DOI10.1007/s00013-004-1227-4
  26. Soullami, A., Sahmoudi, M., Boughaleb, O., 10.1216/rmj.2021.51.1443, Rocky Mt. J. Math. 51 (2021), 1443-1452. (2021) Zbl1469.11414MR4298858DOI10.1216/rmj.2021.51.1443
  27. Spearman, B. K., Williams, K. S., 10.1007/BF02188204, Acta Math. Hung. 70 (1996), 185-192. (1996) MR1374384DOI10.1007/BF02188204
  28. Spearman, B. K., Williams, K. S., 10.1155/s0161171203204336, Int. J. Math. Math. Sci. 25 (2003), 1623-1626. (2003) Zbl1064.11070MR1979698DOI10.1155/s0161171203204336
  29. Washington, L. C., 10.1090/S0002-9939-1976-0399041-9, Proc. Am. Math. Soc. 56 (1976), 93-94. (1976) Zbl0331.12002MR0399041DOI10.1090/S0002-9939-1976-0399041-9

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.