Currently displaying 1 – 6 of 6

Showing per page

Order by Relevance | Title | Year of publication

Associated orders of certain extensions arising from Lubin-Tate formal groups

Nigel P. Byott — 1997

Journal de théorie des nombres de Bordeaux

Let k be a finite extension of p , let k 1 , respectively k 3 , be the division fields of level 1 , respectively 3 , arising from a Lubin-Tate formal group over k , and let Γ = Gal( k 3 / k 1 ). It is known that the valuation ring k 3 cannot be free over its associated order 𝔄 in K Γ unless k = p . We determine explicitly under the hypothesis that the absolute ramification index of k is sufficiently large.

Galois structure of ideals in wildly ramified abelian p -extensions of a p -adic field, and some applications

Nigel P. Byott — 1997

Journal de théorie des nombres de Bordeaux

Let K be a finite extension of p with ramification index e , and let L / K be a finite abelian p -extension with Galois group Γ and ramification index p n . We give a criterion in terms of the ramification numbers t i for a fractional ideal 𝔓 h of the valuation ring S of L not to be free over its associated order 𝔄 ( K Γ ; 𝔓 h ) . In particular, if t n - [ t n / p ] < p n - 1 e then the inverse different can be free over its associated order only when t i - 1 (mod p n ) for all i . We give three consequences of this. Firstly, if 𝔄 ( K Γ ; S ) is a Hopf order and S is 𝔄 ( K Γ ; S ) -Galois...

A valuation criterion for normal basis generators of Hopf-Galois extensions in characteristic p

Nigel P. Byott — 2011

Journal de Théorie des Nombres de Bordeaux

Let S / R be a finite extension of discrete valuation rings of characteristic p > 0 , and suppose that the corresponding extension L / K of fields of fractions is separable and is H -Galois for some K -Hopf algebra H . Let 𝔻 S / R be the different of S / R . We show that if S / R is totally ramified and its degree n is a power of p , then any element ρ of L with v L ( ρ ) - v L ( 𝔻 S / R ) - 1 ( mod n ) generates L as an H -module. This criterion is best possible. These results generalise to the Hopf-Galois situation recent work of G. G. Elder for Galois extensions.

Relative Galois module structure of integers of abelian fields

Nigel P. ByottGünter Lettl — 1996

Journal de théorie des nombres de Bordeaux

Let L / K be an extension of algebraic number fields, where L is abelian over . In this paper we give an explicit description of the associated order 𝒜 L / K of this extension when K is a cyclotomic field, and prove that o L , the ring of integers of L , is then isomorphic to 𝒜 L / K . This generalizes previous results of Leopoldt, Chan Lim and Bley. Furthermore we show that 𝒜 L / K is the maximal order if L / K is a cyclic and totally wildly ramified extension which is linearly disjoint to ( m ' ) / K , where m ' is the conductor of K .

New ramification breaks and additive Galois structure

Nigel P. ByottG. Griffith Elder — 2005

Journal de Théorie des Nombres de Bordeaux

Which invariants of a Galois p -extension of local number fields L / K (residue field of char p , and Galois group G ) determine the structure of the ideals in L as modules over the group ring p [ G ] , p the p -adic integers? We consider this question within the context of elementary abelian extensions, though we also briefly consider cyclic extensions. For elementary abelian groups G , we propose and study a new group (within the group ring 𝔽 q [ G ] where 𝔽 q is the residue field) and its resulting ramification filtrations....

Realizable Galois module classes over the group ring for non abelian extensions

Nigel P. ByottBouchaïb Sodaïgui — 2013

Annales de l’institut Fourier

Given an algebraic number field k and a finite group Γ , we write ( O k [ Γ ] ) for the subset of the locally free classgroup Cl ( O k [ Γ ] ) consisting of the classes of rings of integers O N in tame Galois extensions N / k with Gal ( N / k ) Γ . We determine ( O k [ Γ ] ) , and show it is a subgroup of Cl ( O k [ Γ ] ) by means of a description using a Stickelberger ideal and properties of some cyclic codes, when k contains a root of unity of prime order p and Γ = V C , where V is an elementary abelian group of order p r and C is a cyclic group of order m > 1 acting faithfully on...

Page 1

Download Results (CSV)