Soit un nombre premier. On dit qu’une extension finie, galoisienne, d’un corps de nombres , à groupe de Galois , admet une base normale -entière (-NIB en abrégé) si est libre de rang un sur l’anneau de groupe où désigne l’anneau des -entiers de . Soit une puissance de et une extension cyclique de degré . Lorsque , nous donnons une condition nécessaire et suffisante pour que admette une -NIB (Théorème 3). Lorsque et , nous montrons que admet une -NIB si et seulement si admet -NIB (Théorème 1). Enfin, si divise , nous montrons que la propriété de descente n’est plus vraie en général (Théorème 2).
Let be a prime number. A finite Galois extension of a number field with group has a normal -integral basis (-NIB for short) when is free of rank one over the group ring . Here, is the ring of -integers of . Let be a power of and a cyclic extension of degree . When , we give a necessary and sufficient condition for to have a -NIB (Theorem 3). When and , we show that has a -NIB if and only if has a -NIB (Theorem 1). When divides , we show that this descent property does not hold in general (Theorem 2).
@article{JTNB_2005__17_3_779_0, author = {Ichimura, Humio}, title = {On the ring of $p$-integers of a cyclic $p$-extension over a number field}, journal = {Journal de th\'eorie des nombres de Bordeaux}, pages = {779--786}, publisher = {Universit\'e Bordeaux 1}, volume = {17}, number = {3}, year = {2005}, doi = {10.5802/jtnb.520}, mrnumber = {2212125}, zbl = {1153.11335}, language = {en}, url = {http://www.numdam.org/articles/10.5802/jtnb.520/} }
TY - JOUR AU - Ichimura, Humio TI - On the ring of $p$-integers of a cyclic $p$-extension over a number field JO - Journal de théorie des nombres de Bordeaux PY - 2005 SP - 779 EP - 786 VL - 17 IS - 3 PB - Université Bordeaux 1 UR - http://www.numdam.org/articles/10.5802/jtnb.520/ DO - 10.5802/jtnb.520 LA - en ID - JTNB_2005__17_3_779_0 ER -
%0 Journal Article %A Ichimura, Humio %T On the ring of $p$-integers of a cyclic $p$-extension over a number field %J Journal de théorie des nombres de Bordeaux %D 2005 %P 779-786 %V 17 %N 3 %I Université Bordeaux 1 %U http://www.numdam.org/articles/10.5802/jtnb.520/ %R 10.5802/jtnb.520 %G en %F JTNB_2005__17_3_779_0
Ichimura, Humio. On the ring of $p$-integers of a cyclic $p$-extension over a number field. Journal de théorie des nombres de Bordeaux, Tome 17 (2005) no. 3, pp. 779-786. doi : 10.5802/jtnb.520. http://www.numdam.org/articles/10.5802/jtnb.520/
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