Number Theory
Sums of integral squares in cyclotomic fields
Comptes Rendus. Mathématique, Volume 344 (2007) no. 7, pp. 413-416.

Let $Kn=Q(ζn)$ be the n-th cyclotomic field with $n≢2(mod4)$. Let $On=Z[ζn]$ be the ring of integers of $Kn$ and $Sn$ the set of all elements $α∈On$ which are sums of squares in $On$. Let $gn$ be the smallest positive integer m such that every element in $Sn$ is a sum of m squares in $On$. In this Note, we show that $gn=3$ unless n is odd and the order of 2 in $(Z/nZ)∗$ is odd, in which case $gn=4$.

Soit $Kn$ le n-ième corps cyclotomique, avec $n≢2(mod4)$, $n>1$. Soit $On$ l'anneau des entiers de $Kn$ et soit $Sn$ le sous-ensemble de $On$ formé des éléments qui sont sommes de carrés. Soit $gn$ le plus petit entier $m>0$ tel que tout élément de $Sn$ soit somme de m carrés d'éléments de $On$. Nous montrons que : $gn=3$ si n est divisible par 4 ; $gn=3$ (resp. $gn=4$) si n est impair et si l'ordre de 2 dans le groupe multiplicatif $(Z/nZ)∗$ est pair (resp. impair).

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2007.02.003

Chun-Gang Ji 1, 2; Da-Sheng Wei 3

1 Department of Mathematics, Nanjing University, Nanjing 210093, P.R. China
2 Department of Mathematics, Nanjing Normal University, Nanjing 210097, P.R. China
3 Department of Mathematics, The University of Science and Technology of China, Hefei 230026, P.R. China
@article{CRMATH_2007__344_7_413_0,
author = {Chun-Gang Ji and Da-Sheng Wei},
title = {Sums of integral squares in cyclotomic fields},
journal = {Comptes Rendus. Math\'ematique},
pages = {413--416},
publisher = {Elsevier},
volume = {344},
number = {7},
year = {2007},
doi = {10.1016/j.crma.2007.02.003},
language = {en},
}
TY  - JOUR
AU  - Chun-Gang Ji
AU  - Da-Sheng Wei
TI  - Sums of integral squares in cyclotomic fields
JO  - Comptes Rendus. Mathématique
PY  - 2007
SP  - 413
EP  - 416
VL  - 344
IS  - 7
PB  - Elsevier
DO  - 10.1016/j.crma.2007.02.003
LA  - en
ID  - CRMATH_2007__344_7_413_0
ER  - 
%0 Journal Article
%A Chun-Gang Ji
%A Da-Sheng Wei
%T Sums of integral squares in cyclotomic fields
%J Comptes Rendus. Mathématique
%D 2007
%P 413-416
%V 344
%N 7
%I Elsevier
%R 10.1016/j.crma.2007.02.003
%G en
%F CRMATH_2007__344_7_413_0
Chun-Gang Ji; Da-Sheng Wei. Sums of integral squares in cyclotomic fields. Comptes Rendus. Mathématique, Volume 344 (2007) no. 7, pp. 413-416. doi : 10.1016/j.crma.2007.02.003. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.02.003/

[1] D.R. Estes; J.S. Hsia Exceptional integers of some ternary quadratic forms, Adv. in Math., Volume 45 (1982), pp. 310-318

[2] D.R. Estes; J.S. Hsia Sums of three integer squares in complex quadratic fields, Proc. Amer. Math. Soc., Volume 89 (1983), pp. 211-214

[3] F. Götzky Über eine zahlentheoretische Anwendung von Modulfunktionen einer Veränderlichen, Math. Ann., Volume 100 (1928), pp. 411-437

[4] J.S. Hsia Representations by integral quadratic forms over algebraic number fields, Conference on Quadratic Forms—1976 (Queen's Papers in Pure and Appl. Math.), Volume vol. 46 (1977), pp. 528-537

[5] C.-G. Ji Sums of three integral squares in cyclotomic fields, Bull. Austral. Math. Soc., Volume 68 (2003), pp. 101-106

[6] C.-G. Ji; Y.-H. Wang; F. Xu Sums of three squares over imaginary quadratic fields, Forum Math., Volume 18 (2006), pp. 585-601

[7] H. Maass Über die Darstellung total positiver Zahlen des Körpers $R(5)$ als Summe von drei Quadraten, Abh. Math. Sem. Hansischen Univ., Volume 14 (1941), pp. 185-191

[8] C. Moser Représentation de −1 par une somme de carrés dans certains corps locaux et globaux, et dans certains anneaux d'entiers algébriques, C. R. Acad. Sci. Paris Ser. A-B, Volume 271 (1970), p. A1200-A1203

[9] H. Qin The sum of two squares in a quadratic field, Comm. Algebra, Volume 25 (1997), pp. 177-184

[10] C.L. Siegel Darstellung total positiver Zahlen durch Quadrate, Math. Z., Volume 11 (1921), pp. 246-275

[11] C.L. Siegel Sums of mth powers of algebraic integers, Ann. of Math., Volume 46 (1945), pp. 313-339

Cited by Sources:

This work was partially supported by the Grant No. 10171046 and 10201013 from NNSF of China and Jiangsu planned projects for postdoctoral research funds.

Comments