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.

Let p be a prime number. A finite Galois extension N/F of a number field F with group G has a normal p-integral basis (p-NIB for short) when 𝒪 N is free of rank one over the group ring 𝒪 F [G]. Here, 𝒪 F =𝒪 F [1/p] is the ring of p-integers of F. Let m=p e be a power of p and N/F a cyclic extension of degree m. When ζ m F × , we give a necessary and sufficient condition for N/F to have a p-NIB (Theorem 3). When ζ m F × and p[F(ζ m ):F], we show that N/F has a p-NIB if and only if N(ζ m )/F(ζ m ) has a p-NIB (Theorem 1). When p divides [F(ζ m ):F], we show that this descent property does not hold in general (Theorem 2).

Soit p un nombre premier. On dit qu’une extension finie, galoisienne, N/F d’un corps de nombres F, à groupe de Galois G, admet une base normale p-entière (p-NIB en abrégé) si 𝒪 N est libre de rang un sur l’anneau de groupe 𝒪 F [G]𝒪 F =𝒪 F [1/p] désigne l’anneau des p-entiers de F. Soit m=p e une puissance de p et N/F une extension cyclique de degré m. Lorsque ζ m F × , nous donnons une condition nécessaire et suffisante pour que N/F admette une p-NIB (Théorème 3). Lorsque ζ m F × et p[F(ζ m ):F], nous montrons que N/F admet une p-NIB si et seulement si N(ζ m )/F(ζ m ) admet p-NIB (Théorème 1). Enfin, si p divise [F(ζ m ):F], nous montrons que la propriété de descente n’est plus vraie en général (Théorème 2).

Published online:
DOI: 10.5802/jtnb.520
Humio Ichimura 1

1 Faculty of Science Ibaraki University 2-1-1, Bunkyo, Mito, Ibaraki, 310-8512 Japan
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Humio Ichimura. 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. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.520/

[1] J. Brinkhuis, Normal integral bases and the Spiegelungssatz of Scholz. Acta Arith. 69 (1995), 1–9. | MR: 1310838 | Zbl: 0838.11073

[2] V. Fleckinger, T. Nguyen-Quang-Do, Bases normales, unités et conjecture faible de Leopoldt. Manus. Math. 71 (1991), 183–195. | MR: 1101268 | Zbl: 0732.11063

[3] A. Fröhlich, M. J. Taylor, Algebraic Number Theory. Cambridge Univ. Press, Cambridge, 1991. | MR: 1215934 | Zbl: 0744.11001

[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: 1305289 | Zbl: 0822.11076

[5] C. Greither, Cyclic Galois Extensions of Commutative Rings. Lect. Notes Math. 1534, Springer–Verlag, 1992. | MR: 1222646 | Zbl: 0788.13003

[6] C. Greither, On normal integral bases in ray class fields over imaginary quadratic fields. Acta Arith. 78 (1997), 315–329. | MR: 1438589 | Zbl: 0866.11064

[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: 1980241 | Zbl: 1048.11087

[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: 2027901 | Zbl: 1042.11074

[9] H. Ichimura, Normal integral bases and ray class groups. Acta Arith. 114 (2004), 71–85. | MR: 2067873 | Zbl: 1065.11090

[10] H. Ichimura, H. Sumida, On the Iwasawa invariants of certain real abelian fields. Tohoku J. Math. 49 (1997), 203–215. | MR: 1447182 | Zbl: 0886.11060

[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: 1821183 | Zbl: 0991.11058

[12] I. Kersten, J. Michalicek, On Vandiver’s conjecture and p -extensions of (ζ p ). J. Number Theory 32 (1989), 371–386. | MR: 1006601 | Zbl: 0709.11058

[13] H. Koch, Algebraic Number Theory. Springer, Berlin-Heidelberg-New York, 1997. | MR: 1474965

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