Soit le n-ième corps cyclotomique, avec , . Soit l'anneau des entiers de et soit le sous-ensemble de formé des éléments qui sont sommes de carrés. Soit le plus petit entier tel que tout élément de soit somme de m carrés d'éléments de . Nous montrons que : si n est divisible par 4 ; (resp. ) si n est impair et si l'ordre de 2 dans le groupe multiplicatif est pair (resp. impair).
Let be the n-th cyclotomic field with . Let be the ring of integers of and the set of all elements which are sums of squares in . Let be the smallest positive integer m such that every element in is a sum of m squares in . In this Note, we show that unless n is odd and the order of 2 in is odd, in which case .
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@article{CRMATH_2007__344_7_413_0, author = {Ji, Chun-Gang and Wei, Da-Sheng}, 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}, url = {http://archive.numdam.org/articles/10.1016/j.crma.2007.02.003/} }
TY - JOUR AU - Ji, Chun-Gang AU - Wei, Da-Sheng TI - Sums of integral squares in cyclotomic fields JO - Comptes Rendus. Mathématique PY - 2007 SP - 413 EP - 416 VL - 344 IS - 7 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/j.crma.2007.02.003/ DO - 10.1016/j.crma.2007.02.003 LA - en ID - CRMATH_2007__344_7_413_0 ER -
%0 Journal Article %A Ji, Chun-Gang %A Wei, Da-Sheng %T Sums of integral squares in cyclotomic fields %J Comptes Rendus. Mathématique %D 2007 %P 413-416 %V 344 %N 7 %I Elsevier %U http://archive.numdam.org/articles/10.1016/j.crma.2007.02.003/ %R 10.1016/j.crma.2007.02.003 %G en %F CRMATH_2007__344_7_413_0
Ji, Chun-Gang; Wei, Da-Sheng. Sums of integral squares in cyclotomic fields. Comptes Rendus. Mathématique, Tome 344 (2007) no. 7, pp. 413-416. doi : 10.1016/j.crma.2007.02.003. http://archive.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.