Displaying similar documents to “Some counter-examples in the theory of the Galois module structure of wild extensions”

Euler system for Galois deformations

Tadashi Ochiai (2005)

Annales de l’institut Fourier

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In this paper, we develop the Euler system theory for Galois deformations. By applying this theory to the Beilinson-Kato Euler system for Hida’s nearly ordinary modular deformations, we prove one of the inequalities predicted by the two-variable Iwasawa main conjecture. Our method of the proof of the Euler system theory is based on non-arithmetic specializations. This gives a new simplified proof of the inequality between the characteristic ideal of the Selmer group of a Galois deformation...

The cyclic subfield integer index

Bart de Smit (2000)

Journal de théorie des nombres de Bordeaux

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In this note we consider the index in the ring of integers of an abelian extension of a number field K of the additive subgroup generated by integers which lie in subfields that are cyclic over K . This index is finite, it only depends on the Galois group and the degree of K , and we give an explicit combinatorial formula for it. When generalizing to more general Dedekind domains, a correction term can be needed if there is an inseparable extension of residue fields. We identify this correction...

Galois module structure of ideals in wildly ramified cyclic extensions of degree p 2

Gove Griffith Elder (1995)

Annales de l'institut Fourier

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For L / K , any totally ramified cyclic extension of degree p 2 of local fields which are finite extensions of the field of p -adic numbers, we describe the p [ Gal ( L / K ) ] -module structure of each fractional ideal of L explicitly in terms of the 4 p + 1 indecomposable p [ Gal ( L / K ) ] -modules classified by Heller and Reiner. The exponents are determined only by the invariants of ramification.

Galois module structure of rings of integers

Martin J. Taylor (1980)

Annales de l'institut Fourier

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Let E / F be a Galois extension of number fields with Γ = Gal ( E / F ) and with property that the divisors of ( E : F ) are non-ramified in E / Q . We denote the ring of integers of E by 𝒪 E and we study 𝒪 E as a Z Γ -module. In particular we show that the fourth power of the (locally free) class of 𝒪 E is the trivial class. To obtain this result we use Fröhlich’s description of class groups of modules and his representative for the class of E , together with new determinantal congruences for cyclic group rings and corresponding...

On Galois structure of the integers in cyclic extensions of local number fields

G. Griffith Elder (2002)

Journal de théorie des nombres de Bordeaux

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Let p be a rational prime, K be a finite extension of the field of p -adic numbers, and let L / K be a totally ramified cyclic extension of degree p n . Restrict the first ramification number of L / K to about half of its possible values, b 1 > 1 / 2 · p e 0 / ( p - 1 ) where e 0 denotes the absolute ramification index of K . Under this loose condition, we explicitly determine the p [ G ] -module structure of the ring of integers of L , where p denotes the p -adic integers and G denotes the Galois group Gal ( L / K ) . In the process of determining...