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, p. 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).

@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},
     publisher = {Universit\'e Bordeaux 1},
     volume = {17},
     number = {3},
     year = {2005},
     pages = {779-786},
     doi = {10.5802/jtnb.520},
     zbl = {1153.11335},
     mrnumber = {2212125},
     zbl = {05016587},
     language = {en},
     url = {http://www.numdam.org/item/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://www.numdam.org/item/JTNB_2005__17_3_779_0/

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