Galois modules in a multidimensional local field

The structure of the group of principal units is studied as a Galois module in a weakly ramified extension of a multidimensional local field. A criterion is given for the existence of a normal basis for additive Galois modules in such extensions. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova, AN SSSR, Vol. 198, pp. 5–14, 1991.     

Galois Structure of K-Groups of Rings of Integers

N/Kbe a Galois extension of number fields with finite Galois group G.We describe a new approach for constructing invariants of the G-module structure of the K groups of the ring of integers of N in the Grothendieck group of finitely generated projective Z[G]modules. In various cases we can relate these classes, and their function field counterparts, to the root number class of Fröhlich and Cassou-Noguès.     

The Invariant Subrings of an Azumaya Galois Extension

Let B be an Azumaya Galois extension or a DeMeyer-Kanzaki Galois extension with Galois group G. Equivalent conditions are given for a separable subextension of a Galois extension in the skew group ring B * G being an invariant subring of a subgroup of the Galois group G.AMS Subject Classification (2000): 16S35, 16W20.     

Trace forms of normal extensions over local fields

Let k be a local field of char(k)≠2 and K/k a finite field extension of degree n. Then K can be viewed as a quadratic space of k under the quadratic form T(X) =tr_K/k(x²). The invariants of this form are given in the case when K/k is a Galois extension, except for Galois extensions K/k with k dyadicn divisible by 4 and the 2-Sylowgroups of the Galois group are non-cyclic. Conversely all quadratic forms of a local field k of char(k)≠ 2 which appear as trace forms of Galois extensions of k are determined.     

Chebyshev?s bias in Galois extensions of global function fields

We study Chebyshev?s bias in a finite, possibly nonabelian, Galois extension of global function fields. We show that, when the extension is geometric and satisfies a certain property, called, Linear Independence (LI), the less square elements a conjugacy class of the Galois group has, the more primes there are whose Frobenius conjugacy classes are equal to the conjugacy class. Our results are in line with the previous work of Rubinstein and Sarnak in the number field case and that of the first-named author in the case of polynomial rings over finite fields. We also prove, under LI, the necessary and sufficient conditions for a certain limiting distribution to be symmetric, following the method of Rubinstein and Sarnak. Examples are provided where LI is proved to hold true and is violated. Also, we study the case when the Galois extension is a scalar field extension and describe the complete result of the prime number race in that case.     

Picard-Vessiot theory for real fields

The existence of a Picard-Vessiot extension for a homogeneous linear differential equation has been established when the differential field over which the equation is defined has an algebraically closed field of constants. In this paper, we prove the existence of a Picard-Vessiot extension for a homogeneous linear differential equation defined over a real differential field K with real closed field of constants. We give an adequate definition of the differential Galois group of a Picard-Vessiot extension of a real differential field with real closed field of constants and we prove a Galois correspondence theorem for such a Picard-Vessiot extension.     

Filtration of the group of principal units of a local function field as a Galois module

Let k be the field of formal power series in one variable over a finite field of constants of characteristic p and K/k be a completely ramified extension of degree p. For the group E₁ of principal units of the field K, considered as a Galois module, generators are constructed effectively. The operator structure (as a Galois module) of the subgroups E_m generating a natural filtration of the group E₁ is determined.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 103, pp. 96–99, 1980.     

Arithmetic of quasi-cyclotomic fields

We call a quadratic extension of a cyclotomic field a quasi-cyclotomic field if it is non-abelian Galois over the rational number field. In this paper, we study the arithmetic of any quasi-cyclotomic field, including to determine the ring of integers of it, the decomposition nature of prime numbers in it, and the structure of the Galois group of it over the rational number field. We also describe explicitly all real quasi-cyclotomic fields, namely, the maximal real subfields of quasi-cyclotomic fields which are Galois over the rational number field. It gives a series of totally real fields and CM fields which are non-abelian Galois over the rational number field.     

Generalized differential Galois theory

A Galois theory of differential fields with parameters is developed in a manner that generalizes Kolchin's theory. It is shown that all connected differential algebraic groups are Galois groups of some appropriate differential field extension.

     

Torsion bounds for elliptic curves and Drinfeld modules

We derive upper bounds on the number of L-rational torsion points on a given elliptic curve or Drinfeld module defined over a finitely generated field K, as a function of the degree [L:K]. Our main tool is the adelic openness of the image of Galois representations, due to Serre, Pink and Rütsche. Our approach is to prove a general result for certain Galois modules, which applies simultaneously to elliptic curves and to Drinfeld modules.     

G-Closed fields and imbeddings of quadratic number fields

A field, K, that has no extensions with Galois group isomorphic to G is called G-closed. It is proved that a finite extension of K admits an infinite number of nonisomorphic extensions with Galois group G. A trinomial of degree n is exhibited with Galois group, the symmetric group of degree n, and with prescribed discriminant. This result is used to show that any quadratic extension of an A_n-closed field admits an extension with Galois group A_n.     

Modularity and incremental innovation: the roles of design rules and organizational communication

As little attention has been paid to the relationship between modularity and near decomposability, extant studies have not unveiled the impact of modularity on incremental innovation completely. We argue that the modular structure is a special case of nearly decomposable structure, in which the interdependencies between modules are specified by design rules, and the degree of modularity is defined by the level of near decomposability and the extent to which intermodule dependencies are specified. The results of our simulation experiments show that in the term of near decomposability, the increase of modularity leads to higher innovation advantage in the short term, but effective communication between modules can help systems with moderate and low modularity gain more innovation benefits in the long term; in the aspect of design rules, modularization may restrict the search space of the incremental innovation within each module, but under some conditions the option value of modularity may offset or even exceed the restriction effect of design rules.     

H-SEPARABLE RINGS AND THEIR HOPF-GALOIS EXTENSIONS

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Bit-Parallel Arithmetic Implementations over Finite Fields GF(2 m ) with Reconfigurable Hardware

Galois (or finite) fields are used in a wide number of technical applications, playing an important role in several areas such as cryptographic schemes and algebraic codes, used in modern digital communication systems. Finite field arithmetic must be fast, due to the increasing performance needed by communication systems, so it might be necessary for the implementation of the modules performing arithmetic over Galois fields on semiconductor integrated circuits. Galois field multiplication is the most costly arithmetic operation and different approaches can be used. In this paper, the fundamentals of Galois fields are reviewed and multiplication of finite-field elements using three different representation bases are considered. These three multipliers have been implemented using a bit-parallel architecture over reconfigurable hardware and experimental results are presented to compare the performance of these multipliers.     

Witt rings and the quaternion group

Let F be a formally real field which admits no quaternionic Galois extension. The structure of the Witt ring and the maximal pro-2 Galois group of F are investigated. Received: 3 July 1997 / Revised version: 2 February 1998     

The structure of a partial Galois extension

In this paper, we present an easy way to construct partial Galois extensions; in particular, any direct sum of finitely many Galois extensions forms a partial Galois extension. The idea is inspired by the study of how Galois extensions are embedded in a partial Galois extension via minimal elements in an associated Boolean semigroup.     

On p-Extensions of Fields of Characteristic p

It is proved that a Galois extension of a field of characteristic p is completely determined by its Galois group and the endomorphism of the additive group of the group algebra that corresponds to the raising to the power p.     

On the Cartan Map for Crossed Products and Hopf–Galois Extensions

We study certain aspects of the algebraic K-theory of Hopf–Galois extensions. We show that the Cartan map from K-theory to G-theory of such an extension is a rational isomorphism, provided the ring of coinvariants is regular, the Hopf algebra is finite dimensional and its Cartan map is injective in degree zero. This covers the case of a crossed product of a regular ring with a finite group and has an application to the study of Iwasawa modules.     

On the Iwasawa Theory of p-Adic Lie Extensions

In this paper, the new techniques and results concerning the structure theory of modules over noncommutative Iwasawa algebras are applied to arithmetic: we study Iwasawa modules over p-adic Lie extensions k of number fields k 'up to pseudo-isomorphism'. In particular, a close relationship is revealed between the Selmer group of Abelian varieties, the Galois group of the maximal Abelian unramified p-extension of k as well as the Galois group of the maximal Abelian p-extension unramified outside S where S is a certain finite setof places of k. Moreover, we determine the Galois module structure of local units and other modules arising from Galois cohomology.