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 n2(mod4). Let On=Z[ζn] be the ring of integers of Kn and Sn the set of all elements αOn 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 n2(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).

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Accepted:
Published online:
DOI: 10.1016/j.crma.2007.02.003
Ji, Chun-Gang 1, 2; Wei, Da-Sheng 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
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Ji, Chun-Gang; Wei, Da-Sheng. 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. http://www.numdam.org/articles/10.1016/j.crma.2007.02.003/

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