Sums of integral squares in cyclotomic fields

Chun-Gang Ji; Da-Sheng Wei

doi:10.1016/j.crma.2007.02.003

Number Theory

Sums of integral squares in cyclotomic fields

Presented by: Jean-Pierre Serre

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

Comptes Rendus. Mathématique, Volume 344 (2007) no. 7, pp. 413-416.

Abstract
Résumé

Let $K_{n} = Q (ζ_{n})$ be the n-th cyclotomic field with $n ≢ 2 (mod 4)$ . Let $O_{n} = Z [ζ_{n}]$ be the ring of integers of $K_{n}$ and $S_{n}$ the set of all elements $α \in O_{n}$ which are sums of squares in $O_{n}$ . Let $g_{n}$ be the smallest positive integer m such that every element in $S_{n}$ is a sum of m squares in $O_{n}$ . In this Note, we show that $g_{n} = 3$ unless n is odd and the order of 2 in ${(Z / n Z)}^{*}$ is odd, in which case $g_{n} = 4$ .

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

Received: 2006-09-19
Accepted: 2007-02-07
Published online: 2007-03-21

DOI: 10.1016/j.crma.2007.02.003

Author's affiliations:

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

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     title = {Sums of integral squares in cyclotomic fields},
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     pages = {413--416},
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     language = {en},
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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/

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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.

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