On the necessity of new ramification breaks

Byott, Nigel P.; Elder, G. Griffith

doi:10.1016/j.jnt.2008.07.011

Cited by 5 publications

(9 citation statements)

References 12 publications

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“…The consideration of cyclic p-extensions of higher degree in positive characteristic is still in progress. [44], and obtained a criterion which agrees with the condition found by Miyata for certain Kummer extensions in characteristic 0 [36,133]).…”

Section: Local Galois Module Structure In Equal Characteristicsupporting

confidence: 79%

On the Galois module structure of extensions of local fields

Thomas¹

2010

Publications Mathématiques De Besançon

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Section: Local Galois Module Structure In Equal Characteristicsupporting

confidence: 79%

On the Galois module structure of extensions of local fields

Thomas¹

2010

Publications Mathématiques De Besançon

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“…In [6], it is shown that the ramification group filtration of a wildly ramified prime p is uniquely determined by the p-adic valuation of the discriminant of the field extension L/K, when both the field extension degree and the residue characteristic of p are equal to a prime number. When the Galois group is elementary abelian, the Galois module structure of certain ideals is related to the ramification group filtration, see [3,4,5]. Such a relation is investigated when the Galois group is quaternion [7], hence non-abelian.…”

Section: This Relation Is Close If An Intersection Propertymentioning

confidence: 99%

The ramification group filtrations of certain function field extensions

Castañeda

2015

Pacific J. Math.

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We investigate the ramification group filtration of a Galois extension of function fields, if the Galois group satisfies a certain intersection property. For finite groups, this property is implied by having only elementary abelian Sylow p-subgroups. Note that such groups could be non-abelian. We show how the problem can be reduced to the totally wild ramified case on a p-extension. Our methodology is based on an intimate relationship between the ramification groups of the field extension and those of all degree p sub-extensions. Not only do we confirm that the Hasse-Arf property holds in this setting, but we also prove that the Hasse-Arf divisibility result is the best possible by explicit calculations of the quotients, which are expressed in terms of the different exponents of all those degree p sub-extensions. v ACKNOWLEDGMENTS

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“…Of course there are many seriesα(X) with this property, but for our purposes it will not matter which we choose. Let φ : K 0 → K 0 be the p-Frobenius map and defineα ∆ (X) =α φ (X p ) and l(α) = log(α) − p −1 log(α ∆ ), where log(1 + ψ(X)) = ψ(X) − 1 2 ψ(X) 2 Proof: By the linearity and continuity of the Kummer pairing we may assume that…”

Section: Kummer Theorymentioning

confidence: 99%

“…If i 1 = p 2 b−pb then b * can take any of the values allowed by Theorem 5 in[2]. On the other hand, for a given b * we have eitheri 1 = p 2 b−pb or i 1 = b * +p 2 b−pb−b.…”

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confidence: 99%

Indices of inseparability and refined ramification breaks

Keating

2014

Journal of Number Theory

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Let K be a finite extension of Q p which contains a primitive pth root of unity ζ p . Let L/K be a totally ramified (Z/pZ) 2 -extension which has a single ramification break b. In [2] Byott and Elder defined a "refined ramification break" b * for L/K. In this paper we prove that if p > 2 and the index of inseparability i 1 of L/K is not equal to p 2 b − pb then b * = i 1 − p 2 b + pb + b.In [1, 2], Byott and Elder described an alternative method for supplying missing ramification data by defining refined lower ramification breaks for extensions with fewer than n ordinary breaks. Suppose L/K is a totally ramified (Z/pZ) 2 -extension with a single (ordinary) ramification break b. Then L/K has one refined break b * , which is computed in [2] under the assumption that K contains a primitive pth root of unity. Byott and Elder also showed that the Galois module structure of O L determines b * in certain cases.In this paper we study the relationship between the index of inseparability i 1 of L/K and the refined ramification break b * . In particular, when p > 2 and i 1 = p 2 b − pb we give a formula which expresses b * in terms of i 1 . Our approach is based on the methods given in [8] for computing i 1 in terms of the norm group N L/K (L × ). We relate these methods to the Byott-Elder formula for b * using Vostokov's formula [9] for computing the Kummer pairing , p : K × × K × → µ p . The calculations are simplified somewhat through the use of the Artin-Hasse exponential series E p (X).The author would like to thank the referee for writing a very careful and thorough review of this paper.

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On the necessity of new ramification breaks

Cited by 5 publications

References 12 publications

On the Galois module structure of extensions of local fields

On the Galois module structure of extensions of local fields

The ramification group filtrations of certain function field extensions

Indices of inseparability and refined ramification breaks

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