Multiply monogenic orders
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 12 (2013) no. 2, p. 467-497

Let A=[x 1 ,...,x r ] be a domain which is finitely generated over and integrally closed in its quotient field L. Further, let K be a finite extension field of L. An A-order in K is a domain 𝒪A with quotient field K which is integral over A. A-orders in K of the type A[α] are called monogenic. It was proved by Győry [10] that for any given A-order 𝒪 in K there are at most finitely many A-equivalence classes of α𝒪 with A[α]=𝒪, where two elements α,β of 𝒪 are called A-equivalent if β=uα+a for some uA * , aA. If the number of A-equivalence classes of α with A[α]=𝒪 is at least k, we call 𝒪 k times monogenic.

In this paper we study orders which are more than one time monogenic. Our first main result is that if K is any finite extension of L of degree 3, then there are only finitely many three times monogenic A-orders in K. Next, we define two special types of two times monogenic A-orders, and show that there are extensions K which have infinitely many orders of these types. Then under certain conditions imposed on the Galois group of the normal closure of K over L, we prove that K has only finitely many two times monogenic A-orders which are not of these types. Some immediate applications to canonical number systems are also mentioned.

Published online : 2019-02-21
Classification:  11R99,  11D99,  11J99
@article{ASNSP_2013_5_12_2_467_0,
     author = {B\'erczes, Attila and Evertse, Jan-Hendrik and Gy\H ory, K\'alm\'an},
     title = {Multiply monogenic orders},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 12},
     number = {2},
     year = {2013},
     pages = {467-497},
     zbl = {1319.11070},
     mrnumber = {3114010},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2013_5_12_2_467_0}
}
Bérczes, Attila; Evertse, Jan-Hendrik; Győry, Kálmán. Multiply monogenic orders. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 12 (2013) no. 2, pp. 467-497. http://www.numdam.org/item/ASNSP_2013_5_12_2_467_0/

[1] A. Bérczes, On the number of solutions of index form equations, Publ. Math. Debrecen 56 (2000), 251–262. | MR 1765979 | Zbl 0961.11010

[2] H. Brunotte, A. Huszti and A. Pethő, Bases of canonical number systems in quartic algebraic number fields, J. Théor. Nombres Bordeaux 18 (2006), 537–557. | Numdam | MR 2330426 | Zbl 1193.11099

[3] N. Bourbaki, “Commutative Algebra”, Chapters 1–7, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 1989. | MR 979760 | Zbl 0673.00001

[4] R. Dedekind, Über die Zusammenhang zwischen der Theorie der Ideale und der Theorie der höheren Kongruenzen, Abh. König. Ges. Wissen. Göttingen 23 (1878), 1–23.

[5] J.-H. Evertse and K. Győry, On unit equations and decomposable form equations, J. Reine Angew. Math. 358 (1985), 6–19. | MR 797671 | Zbl 0552.10010

[6] J.-H. Evertse, K. Győry, C. L. Stewart and R. Tijdeman, On S-unit equations in two unknowns, Invent. Math. 92 (1988), 461–477. | MR 939471 | Zbl 0662.10012

[7] K. Győry, Sur les polynômes à coefficients entiers et de discriminant donné, Acta Arith. 23 (1973), 419–426. | MR 437489 | Zbl 0269.12001

[8] K. Győry, Sur les polynômes à coefficients entiers et de dicriminant donné III, Publ. Math. Debrecen 23 (1976), 141–165. | MR 437491 | Zbl 0354.10041

[9] K. Győry, Corps de nombres algébriques d’anneau d’entiers monogène, In: “Séminaire Delange-Pisot-Poitou”, 20e année: 1978/1979. Théorie des nombres, Fasc. 2 (French), Secrétariat Math., Paris, 1980, pp. Exp. No. 26, 7. | Numdam | MR 582432 | Zbl 0433.12001

[10] K. Győry, On certain graphs associated with an integral domain and their applications to Diophantine problems, Publ. Math. Debrecen 29 (1982), 79–94. | MR 673141 | Zbl 0522.10013

[11] K. Győry, Effective finiteness theorems for polynomials with given discriminant and integral elements with given discriminant over finitely generated domains, J. Reine Angew. Math. 346 (1984), 54–100. | MR 727397 | Zbl 0519.13008

[12] K. Győry, Upper bounds for the number of solutions of unit equations in two unknowns, Lithuanian Math. J. 32 (1992), 40–44. | MR 1206381 | Zbl 0814.11018

[13] K. Győry, Polynomials and binary forms with given discriminant, Publ. Math. Debrecen 69 (2006), 473–499. | MR 2274970 | Zbl 1121.11073

[14] L.-C. Kappe and B. Warren, An elementary test for the Galois group of a quartic polynomial, Amer. Math. Monthly 96 (1989), 133–137. | MR 992075 | Zbl 0702.11075

[15] B. Kovács, Canonical number systems in algebraic number fields, Acta Math. Acad. Sci. Hungar. 37 (1981), 405–407. | MR 619892 | Zbl 0505.12001

[16] B. Kovács and A. Pethő, Number systems in integral domains, especially in orders of algebraic number fields, Acta Sci. Math. 55 (1991), 287–299. | MR 1152592 | Zbl 0760.11002

[17] S. Lang, Integral points on curves, Inst. Hautes Études Sci. Publ. Math. 6 (1960), 27–43. | Numdam | MR 130219 | Zbl 0112.13402

[18] M. Laurent, Équations diophantiennes exponentielles, Invent. Math. 78 (1984), 299–327. | MR 767195 | Zbl 0554.10009

[19] P. Roquette, Einheiten und Divisorklassen in endlich erzeugbaren Körpern, Jber. Deutsch. Math. Verein 60 (1957), 1–21. | MR 104652 | Zbl 0079.26901

[20] B.L. van der Waerden, “Algebra I” (8. Auflage), Springer Verlag, 1971. | MR 177027 | Zbl 0221.12001