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).
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).
@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://archive.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://archive.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://archive.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, Volume 17 (2005) no. 3, pp. 779-786. doi : 10.5802/jtnb.520. http://archive.numdam.org/articles/10.5802/jtnb.520/
[1] J. Brinkhuis, Normal integral bases and the Spiegelungssatz of Scholz. Acta Arith. 69 (1995), 1–9. | MR | Zbl
[2] V. Fleckinger, T. Nguyen-Quang-Do, Bases normales, unités et conjecture faible de Leopoldt. Manus. Math. 71 (1991), 183–195. | MR | Zbl
[3] A. Fröhlich, M. J. Taylor, Algebraic Number Theory. Cambridge Univ. Press, Cambridge, 1991. | MR | Zbl
[4] E. J. Gómez Ayala, Bases normales d’entiers dans les extensions de Kummer de degré premier. J. Théor. Nombres Bordeaux 6 (1994), 95–116. | Numdam | MR | Zbl
[5] C. Greither, Cyclic Galois Extensions of Commutative Rings. Lect. Notes Math. 1534, Springer–Verlag, 1992. | MR | Zbl
[6] C. Greither, On normal integral bases in ray class fields over imaginary quadratic fields. Acta Arith. 78 (1997), 315–329. | MR | Zbl
[7] H. Ichimura, On a theorem of Childs on normal bases of rings of integers. J. London Math. Soc. (2) 68 (2003), 25–36: Addendum. ibid. 69 (2004), 303–305. | MR | Zbl
[8] H. Ichimura, On the ring of integers of a tame Kummer extension over a number field. J. Pure Appl. Algebra 187 (2004), 169–182. | MR | Zbl
[9] H. Ichimura, Normal integral bases and ray class groups. Acta Arith. 114 (2004), 71–85. | MR | Zbl
[10] H. Ichimura, H. Sumida, On the Iwasawa invariants of certain real abelian fields. Tohoku J. Math. 49 (1997), 203–215. | MR | Zbl
[11] H. Ichimura, H. Sumida, A note on integral bases of unramified cyclic extensions of prime degree, II. Manus. Math. 104 (2001), 201–210. | MR | Zbl
[12] I. Kersten, J. Michalicek, On Vandiver’s conjecture and -extensions of . J. Number Theory 32 (1989), 371–386. | MR | Zbl
[13] H. Koch, Algebraic Number Theory. Springer, Berlin-Heidelberg-New York, 1997. | MR
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