no. 3
Cubic polynomials defining monogenic fields with the same discriminant
Davis, Chad T.1; Spearman, Blair K.; Yoo, Jeewon2
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
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 =ℤ[θ] for some θ∈𝒪 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 =ℤ[θ] pour un certain θ∈𝒪 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
Davis, Chad T. 1; Spearman, Blair K. ; Yoo, Jeewon 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 = {Davis, Chad T. and Spearman, Blair K. and Yoo, Jeewon},
     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},
     mrnumber = {3938638},
     zbl = {07081584},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/jtnb.1061/}
}
TY  - JOUR
AU  - Davis, Chad T.
AU  - Spearman, Blair K.
AU  - Yoo, Jeewon
TI  - Cubic polynomials defining monogenic fields with the same discriminant
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2018
SP  - 991
EP  - 996
VL  - 30
IS  - 3
PB  - Société Arithmétique de Bordeaux
UR  - http://archive.numdam.org/articles/10.5802/jtnb.1061/
DO  - 10.5802/jtnb.1061
LA  - en
ID  - JTNB_2018__30_3_991_0
ER  - 
%0 Journal Article
%A Davis, Chad T.
%A Spearman, Blair K.
%A Yoo, Jeewon
%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 http://archive.numdam.org/articles/10.5802/jtnb.1061/
%R 10.5802/jtnb.1061
%G en
%F JTNB_2018__30_3_991_0
Davis, Chad T.; Spearman, Blair K.; Yoo, Jeewon. 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. http://archive.numdam.org/articles/10.5802/jtnb.1061/

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

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

[3] Llorente, Pascual; Nart, Enric 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 | DOI | MR | Zbl

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

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

[6] Stewart, Cameron L.; Top, Jaap 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 | DOI | MR | Zbl

Cited by Sources: