Cubic polynomials defining monogenic fields with the same discriminant
Journal de Théorie des Nombres de Bordeaux, Volume 30 (2018) no. 3, pp. 991-996.

Let K be a number field with ring of integers 𝒪 K . K is said to be monogenic if 𝒪 K =[θ] for some θ𝒪 K . Monogeneity of a number field is not always guaranteed. Furthermore, it is rare for two number fields to have the same discriminant, thus finding fields with these two properties is an interesting problem. In this paper we show that there exist infinitely many triples of polynomials defining distinct monogenic cubic fields with the same discriminant.

Un corps de nombres K est dit monogène si son anneau des entiers vérifie 𝒪 K =[θ] pour un certain θ𝒪 K . La monogénéité d’un corps de nombres n’est pas toujours assurée. En outre, il est rare que deux corps de nombres aient le même discriminant. Donc, trouver des corps avec ces deux propriétés est un problème intéressant. Dans cet article, nous montrons qu’il existe une infinité de triplets de polynômes définissant des corps cubiques monogènes distincts de même discriminant.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.1061
Classification: 11R16,  11R29
Keywords: Cubic field, monogenic, discriminant
Chad T. Davis 1; Blair K. Spearman ; Jeewon Yoo 2

1 3333 University Way University of British Columbia - Okanagan Kelowna, BC, Canada, V1V 1V7.
2 3333 University Way University of British Columbia - Okanagan Kelowna, BC, Canada, V1V 1V7
@article{JTNB_2018__30_3_991_0,
     author = {Chad T. Davis and Blair K. Spearman and Jeewon Yoo},
     title = {Cubic polynomials defining monogenic fields with the same discriminant},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {991--996},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {30},
     number = {3},
     year = {2018},
     doi = {10.5802/jtnb.1061},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1061/}
}
TY  - JOUR
TI  - Cubic polynomials defining monogenic fields with the same discriminant
JO  - Journal de Théorie des Nombres de Bordeaux
PY  - 2018
DA  - 2018///
SP  - 991
EP  - 996
VL  - 30
IS  - 3
PB  - Société Arithmétique de Bordeaux
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1061/
UR  - https://doi.org/10.5802/jtnb.1061
DO  - 10.5802/jtnb.1061
LA  - en
ID  - JTNB_2018__30_3_991_0
ER  - 
%0 Journal Article
%T Cubic polynomials defining monogenic fields with the same discriminant
%J Journal de Théorie des Nombres de Bordeaux
%D 2018
%P 991-996
%V 30
%N 3
%I Société Arithmétique de Bordeaux
%U https://doi.org/10.5802/jtnb.1061
%R 10.5802/jtnb.1061
%G en
%F JTNB_2018__30_3_991_0
Chad T. Davis; Blair K. Spearman; Jeewon Yoo. Cubic polynomials defining monogenic fields with the same discriminant. Journal de Théorie des Nombres de Bordeaux, Volume 30 (2018) no. 3, pp. 991-996. doi : 10.5802/jtnb.1061. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1061/

[1] Saban Alaca p-integral bases of a cubic field, Proc. Am. Math. Soc., Volume 126 (1998) no. 7, pp. 1949-1953 | Zbl: 0908.11048

[2] Zenon I. Borevich; Igor R. Shafarevich Number Theory, Academic Press Inc., 1966 | Zbl: 0145.04902

[3] Pascual Llorente; Enric Nart Effective determination of the decomposition of the rational primes in a cubic field, Proc. Am. Math. Soc., Volume 87 (1983) no. 4, pp. 579-585 | Zbl: 0514.12003

[4] Daniel C. Mayer How many fields share a common discriminant? (Multiplicity problem) (Algebra and Algebraic Number Theory, http://www.algebra.at/index_e.htm)

[5] Richard A. Mollin Algebraic Number Theory, Discrete Mathematics and its Applications, CRC Press, 2011 | Zbl: 1219.11001

[6] Cameron L. Stewart; Jaap Top On ranks of twists of elliptic curves and power-free values of binary forms, J. Am. Math. Soc., Volume 8 (1995) no. 4, pp. 943-973 | Zbl: 0857.11026

Cited by Sources: