Bases normales, unités et conjecture faible de leopoldt

Fleckinger, Vincent; Quang, Thong Nguyen

doi:10.1007/bf02568401

Cited by 10 publications

(4 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…These results for the unramified case form a basis of the study of a normal p-integral basis problem for Zp-extensions in Kersten and Michalicek [12], [5] and Fleckinger and Nguyen-Quang-Do [2]. The purpose of this article is to give some corresponding results for the ramified case.…”

Section: Introductionmentioning

confidence: 83%

On the ring of p-integers of a cyclic p-extension over a number field

Ichimura¹

2005

J. Théor. Nombres Bordeaux

View full text Add to dashboard Cite

On the ring of p-integers of a cyclic p-extension over a number field par HUMIO ICHIMURA RÉSUMÉ. 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-entiere (p-NIB en abrégé) si O'N est libre de rang un sur l'anneau de groupe O'F[G] où O'F = O'F[1/p] désigne l'anneau des p-entiers de F. Soit m = pe une puissance de p et N/F une extension cyclique de degré m. Lorsque 03B6m ~ F , nous donnons une condition nécessaire et suffisante pour que N/F admette une p-NIB (Théorème 3). Lorsque 03B6m ~-F et p ~ [F(03B6m) : F], nous montrons que N/F admet une p-NIB si et seulement si N(03B6m)/F(03B6m) admet p-NIB (Théorème 1). Enfin, si p divise [F(03B6m): F], nous montrons que la propriété de descente n'est plus vraie en général (Théorème 2). ABSTRACT. 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 = OF[1/p] is the ring of p-integers of F. Let m = pe be a power of p and N/F a cyclic extension of degree m. When 03B6m ~ Fx, we give a necessary and sufficient condition for N/F to have a p-NIB (Theorem 3). When 03B6m ~ F and p ~ [F(03B6m) : F], we show that N/F has a p-NIB if and only if N(03B6m)/F(03B6m) has a p-NIB (Theorem 1). When p divides [F(03B6m) : F], we show that this descent property does not hold in general (Theorem 2).

show abstract

Section: Introductionmentioning

confidence: 83%

On the ring of p-integers of a cyclic p-extension over a number field

Ichimura¹

2005

J. Théor. Nombres Bordeaux

View full text Add to dashboard Cite

show abstract

“…It was shown in [7] and [8] that if 3 divides the class number of L and if every Z 3 -extension of L 0 has a normal basis, then the -invariant of the cyclotomic Z 3 -extension of L does not vanish. This suggests a relation between the existence of normal basis in Z p -extension and the Greenberg's conjecture, which motivates this section.…”

Section: Construction Of Normal Basesmentioning

confidence: 99%

On some arithmetic properties of Siegel functions (II)

2014

View full text Add to dashboard Cite

Let K be an imaginary quadratic field of discriminant d K Ä 7. We deal with problems of constructing normal bases between abelian extensions of K by making use of singular values of Siegel functions. First, we find normal bases of ring class fields of orders of bounded conductors depending on d K over K by using a criterion deduced from the Frobenius determinant relation. Next, denoting by K .N / the ray class field modulo N of K for an integer N 2 we consider the field extension K .p 2 m/ =K .pm/ for a prime p 5 and a positive integer m relatively prime to p and then find normal bases of all intermediate fields over K .pm/ by utilizing Kawamoto's arguments. We further investigate certain Galois module structure of the field extension K .p n m/ =K .p`m/ with n 2`, which would be an extension of Komatsu's work.

show abstract

“…This paper is organized as follows. In Section 2, we introduce a filtration (see (4)) on the group of semi-local units of K n at p related to some p-adic behaviour of global units, and reduce the questions on T 0 n and N 0 n to questions on this filtration. In Section 3, we first recall some fundamental results on local units and local units modulo cyclotomic units due to Iwasawa and Gillard.…”

Section: Introductionmentioning

confidence: 99%

“…[27], Replogle [28]. (II) The connection between normal integral basis problems and cyclotomic Iwasawa theory is pursued not only in our papers but also in [4,11,25,29], etc. This paper is organized as follows.…”

Section: Introductionmentioning

confidence: 99%

On a Normal Integral Basis Problem over Cyclotomic Z-extensions, II

Ichimura¹

2002

Journal of Number Theory

View full text Add to dashboard Cite

Bases normales, unités et conjecture faible de leopoldt

Cited by 10 publications

References 8 publications

On the ring of p-integers of a cyclic p-extension over a number field

On the ring of p-integers of a cyclic p-extension over a number field

On some arithmetic properties of Siegel functions (II)

On a Normal Integral Basis Problem over Cyclotomic Z-extensions, II

Contact Info

Product

Resources

About