Artin–Schreier extensions and generalized associated orders

Huynh, Duc Van

doi:10.1016/j.jnt.2013.09.009

Cited by 4 publications

(5 citation statements)

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“…(3) The inverse different D −1 L/K is free over its associated order if and only if b = p − 1. These statements also follow from [Fer73,Aib03,dST07,Mar13,Huy14]. The purpose of this paper is to extend these cyclic results to typical extensions L/K, including those that are not Galois.…”

Section: Introductionsupporting

confidence: 57%

“…Indeed, (4) determine the number of generators for P n L over A H (n) if it is not free, and (5) determine the embedding dimension dim κ (M/M 2 ). Indeed, as a result of the scaffold, the results of [Fer73,Aib03,dST07,Mar13,Huy14] under (6), proven for Galois extensions, hold for non-Galois extensions as well.…”

Section: Hopf-galois Module Structurementioning

confidence: 95%

“…For general extensions, this and variations of this question present very difficult problems, and progress starting in the 1970s has been slow. On the other hand, for one specific class of extensions, cyclic of degree p, progress has been good [Fer73,Aib03,dST07,Mar13,Huy14]. One explanation for this progress is that cyclic ramified extensions of degree p naturally possess a scaffold.…”

Section: Introductionmentioning

confidence: 99%

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Ramified extensions of degree p and their Hopf–Galois module structure

Elder¹

2018

J. Théor. Nombres Bordeaux

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Cyclic, ramified extensions L/K of degree p of local fields with residue characteristic p are fairly well understood. Unless char(K) = 0 and L = K( p √ π K ) for some prime element π K ∈ K, they are defined by an Artin-Schreier equation. Additionally, through the work of Ferton, Aiba, de Smit and Thomas, and others, much is known about their Galois module structure of ideals, the structure of each ideal P n L as a module over its associated order A K[G] (n) = {x ∈ K[G] : xP n L ⊆ P n L } where G = Gal(L/K). This paper extends these results to separable, ramified extensions of degree p that are not Galois.

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Section: Introductionsupporting

confidence: 57%

Section: Hopf-galois Module Structurementioning

confidence: 95%

Section: Introductionmentioning

confidence: 99%

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Ramified extensions of degree p and their Hopf–Galois module structure

Elder¹

2018

J. Théor. Nombres Bordeaux

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“…In characteristic 0, Ferton [Fer73] determines which ideals are free over their associated orders, giving her result in terms of the continued fraction expansion of b 1 /p. A corresponding result in characteristic p is given by Huynh [Huy14], who gives a different criterion but proves it is equivalent to Ferton's. Our condition, w(s) = d(s) for all s, must therefore be equivalent to Ferton's continued fraction criterion.…”

Section: Thus We Havementioning

confidence: 99%

Scaffolds and generalized integral Galois module structure

Byott¹,

Childs²,

Elder³

2018

Ann. inst. Fourier

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Let L/K be a finite, totally ramified p-extension of complete local fields with residue fields of characteristic p > 0, and let A be a K-algebra acting on L. We define the concept of an A-scaffold on L, thereby extending and refining the notion of a Galois scaffold considered in several previous papers, where L/K was Galois and A = K[G] for G = Gal(L/K). When a suitable A-scaffold exists, we show how to answer questions generalizing those of classical integral Galois module theory. We give a necessary and sufficient condition, involving only numerical parameters, for a given fractional ideal to be free over its associated order in A. We also show how to determine the number of generators required when it is not free, along with the embedding dimension of the associated order. In the Galois case, the numerical parameters are the ramification breaks associated with L/K. We apply these results to biquadratic Galois extensions in characteristic 2, and to totally and weakly ramified Galois p-extensions in characteristic p. We also apply our results to the non-classical situation where L/K is a finite primitive purely inseparable extension of arbitrary exponent that is acted on, via a higher derivation (but in many different ways), by the divided power K-Hopf algebra.

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“…Example 1.2 in itself is nothing new. Indeed, far more comprehensive treatments of the valuation ring of an extension of degree p are given in [BF72,BBF72] for the characteristic 0 case, and in [Aib03,dST07] for characteristic p. (See also [Fer73] for arbitrary ideals in characteristic 0, and [Huy14] and [Mar14] for the corresponding problem in characteristic p.) We now consider what happens if we try to make the same argument for a larger extension.…”

Section: Introductionmentioning

confidence: 99%

Sufficient conditions for large Galois scaffolds

Byott

Elder

2018

Journal of Number Theory

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Let L/K be a finite, Galois, totally ramified p-extension of complete local fields with perfect residue fields of characteristic p > 0. In this paper, we give conditions, valid for any Galois p-group G = Gal(L/K) (abelian or not) and for K of either possible characteristic (0 or p), that are sufficient for the existence of a Galois scaffold. The existence of a Galois scaffold makes it possible to address questions of integral Galois module structure, which is done in a separate paper [BCE]. But since our conditions can be difficult to check, we specialize to elementary abelian extensions and extend the main result of [Eld09] from characteristic p to characteristic 0. This result is then applied, using a result of Bondarko, to the construction of new Hopf orders over the valuation ring O K that lie in K[G] for G an elementary abelian p-group.

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Artin–Schreier extensions and generalized associated orders

Cited by 4 publications

References 5 publications

Ramified extensions of degree p and their Hopf–Galois module structure

Ramified extensions of degree p and their Hopf–Galois module structure

Scaffolds and generalized integral Galois module structure

Sufficient conditions for large Galois scaffolds

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