Factorisability and the arithmetic of wildly ramified Galois extensions

D. J. Burns

Displaying similar documents to “Factorisability and the arithmetic of wildly ramified Galois extensions”

Factorisability and wildly ramified Galois extensions

David J. Burns (1991)

Annales de l'institut Fourier

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Let $L / K$ be an abelian extension of $p$ -adic fields, and let $𝒪$ denote the valuation ring of $K$ . We study ideals of the valuation ring of $L$ as integral representations of the Galois group $Gal (L / K)$ . Assuming $K$ is absolutely unramified we use techniques from the theory of factorisability to investigate which ideals are isomorphic to an $𝒪$ -order in the group algebra $K [Gal (l / K)]$ . We obtain several general and also explicit new results.

On Galois cohomology and realizability of 2-groups as Galois groups II

Ivo Michailov (2011)

Open Mathematics

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In [Michailov I.M., On Galois cohomology and realizability of 2-groups as Galois groups, Cent. Eur. J. Math., 2011, 9(2), 403–419] we calculated the obstructions to the realizability as Galois groups of 14 non-abelian groups of order 2n, n ≥ 4, having a cyclic subgroup of order 2n−2, over fields containing a primitive 2n−3th root of unity. In the present paper we obtain necessary and sufficient conditions for the realizability of the remaining 8 groups that are not direct products of...

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...

Some results on local fields

Akram Lbekkouri (2013)

Annales UMCS, Mathematica

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Let K be a local field with finite residue field of characteristic p. This paper is devoted to the study of the maximal abelian extension of K of exponent p−1 and its maximal p-abelian extension, especially the description of their Galois groups in solvable case. Then some properties of local fields in general case are studied too.

On Galois cohomology and realizability of 2-groups as Galois groups

Ivo Michailov (2011)

Open Mathematics

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In this paper we develop some new theoretical criteria for the realizability of p-groups as Galois groups over arbitrary fields. We provide necessary and sufficient conditions for the realizability of 14 of the 22 non-abelian 2-groups having a cyclic subgroup of index 4 that are not direct products of groups.

On the Galois structure of the square root of the codifferent

D. Burns (1991)

Journal de théorie des nombres de Bordeaux

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Let $L$ be a finite abelian extension of $ℚ$ , with $𝒪_{L}$ the ring of algebraic integers of $L$ . We investigate the Galois structure of the unique fractional $𝒪_{L}$ -ideal which (if it exists) is unimodular with respect to the trace form of $L / ℚ$ .

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 $ℰ_{E}$ , together with new determinantal congruences for cyclic group rings and corresponding...

Galois module structure of Milnor $K$ -theory mod $p^{s}$ in characteristic $p$ .

Mináč, Ján, Schultz, Andrew, Swallow, John (2008)

The New York Journal of Mathematics [electronic only]

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Galois module structure of generalized jacobians.

G. D. Villa-Salvador, M. Rzedowski-Calderón (1997)

Revista Matemática de la Universidad Complutense de Madrid

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For a prime number l and for a finite Galois l-extension of function fields L / K over an algebraically closed field of characteristic p <> l, it is obtained the Galois module structure of the generalized Jacobian associated to L, l and the ramified prime divisors. In the cyclic case an implicit integral representation of the Jacobian is obtained and this representation is compared with the explicit representation.