GivenS|R a finite Galois extension of finite chain rings andB anS-linear code we define twoGalois operators, the closure operator and the interior operator. We proof that a linear code is Galois invariant if and only if the row standard form of its generator matrix has all entries in the fixed ring by the Galois group and show a Galois correspondence in the class of S-linear codes. As applicati...

We describe a simplified categorical approach to Galois descent theory. It is well known that Galois descent is a special case of Grothendieck descent, and that under mild additional conditions the category of Grothendieck descent data coincides with the Eilenberg-Moore category of algebras over a suitable monad. This also suggests using monads directly, and our monadic approach to Galois desce...

Let S be the left R-bialgebroid of a depth two extension with cen-tralizer R. We show that the left endomorphism ring of depth two extension, not necessarily balanced, is a left S-Galois extension of A op. Looking to examples of depth two, we establish that a Hopf subalgebra is normal if and only if it is a Hopf-Galois extension. We also characterize weak Hopf-Galois extensions using an alterna...

A Galois correspondence for finitely generated field extensions k/h is presented in the case characteristic h = p ^ 0. A field extension k/h is Galois if it is modular and h is separably algebraically closed in k. Galois groups are the direct limit of groups of higher derivations having rank a power of p. Galois groups are characterized in terms of abelian iterative generating sets in a manner ...

We describe Galois extensions where the Galois group is the quasidihedral, dihedral or modular group of order 16, and use this description to produce generic polynomials.

Software-based Galois field implementations are used in the reliability and security components of many storage systems. Unfortunately, multiplication and division operations over Galois fields are expensive, compared to the addition. To accelerate multiplication and division, most software Galois field implementations use pre-computed look-up tables, accepting the memory overhead associated wi...

10. Galois modules and class field theory Boas Erez In this section we shall try to present the reader with a sample of several significant instances where, on the way to proving results in Galois module theory, one is lead to use class field theory. Conversely, some contributions of Galois module theory to class fields theory are hinted at. We shall also single out some problems that in our op...

Absolute Galois Group defined as Galois group of algebraic numbers regarded as extension of rationals is very difficult concept to define. The goal of classical Langlands program is to understand the Galois group of algebraic numbers as algebraic extension of rationals Absolute Galois Group (AGG) through its representations. Invertible adeles -ideles define Gl1 which can be shown to be isomorph...

We discuss some of the basic ideas of Galois theory for commutative S-algebras originally formulated by John Rognes. We restrict attention to the case of finite Galois groups and to global Galois extensions. We describe parts of the general framework developed by Rognes. Central rôles are played by the notion of strong duality and a trace mapping constructed by Greenlees and May in the context ...

Our basic representation of the data is a Galois lattice, i.e. a lattice in which the terms of a representation language are partitioned into equivalence classes w.r.t. their extent (the extent of a term is the part of the instance set that satisfies the term). We propose here to simplify our view of the data, still conserving the Galois lattice formal structure. For that purpose we use a preli...