4 Introduction to Galois Theory 2 4.1 Polynomial Rings . . . . . . . . . . . . . . . . . . . . . . . . . 2 4.2 Gauss’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . 5 4.3 Eisenstein’s Irreducibility Criterion . . . . . . . . . . . . . . . 6 4.4 Field Extensions and the Tower Law . . . . . . . . . . . . . . 6 4.5 Algebraic Field Extensions . . . . . . . . . . . . . . . . . . . . 8 4....

We describe 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. We describe the general framework developed by Rognes. Central rôles are played by the notion of strong duality and a trace or norm mapping constructed by Greenlees and May in the context of generalized Tate cohomology. ...

In this paper we present a reformulation of the Galois correspondence theorem of Hopf Galois theory in terms of groups carrying farther the description of Greither and Pareigis. We prove that the class of Hopf Galois extensions for which the Galois correspondence is bijective is larger than the class of almost classically Galois extensions but not equal to the whole class. We show as well that ...

Galois groups of infinite p-extensions of number fields unramified at p are a complete mystery. We find by computer a family of pro-p groups that satisfy everything that such a Galois group must, and give evidence for the conjecture that these are the only such groups. This suggests that these mysterious Galois groups indeed have a specific form of presentation. There are surprising connections...

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...

This paper primarily concerns Galois cohomology groups associated to Galois representations over a complete local ring R. The underlying Galois module and the corresponding cohomology groups which we consider are discrete R-modules. Under certain hypotheses, we prove that the first cohomology group is an almost divisible Rmodule. We also consider the subgroup of locally trivial elements in the ...

We present a detailed and simplified version of Hrushovski’s algorithm that determines the Galois group of a linear differential equation. There are three major ingredients in this algorithm. The first is to look for a degree bound for proto-Galois groups, which enables one to compute one of them. The second is to determine the identity component of the Galois group that is the pullback of a to...

This paper is concerned with difference equations on elliptic curves. We establish some general properties of the difference Galois groups of equations of order two, and give applications to the calculation of some difference Galois groups. For instance, our results combined with a result from transcendence theory due to Schneider allow us to identify a large class of discrete Lamé equations wi...

in a previous paper, the second author established that, given finite fields $f < e$ and certain subgroups $c leq e^times$, there is a galois connection between the intermediate field lattice ${l mid f leq l leq e}$ and $c$'s subgroup lattice. based on the galois connection, the paper then calculated the irreducible, complex character degrees of the semi-direct product $c rtimes {rm gal} (e/f)$...

We prove many new cases of the Inverse Galois Problem for those simple groups arising from orthogonal groups over finite fields. For example, we show that the finite simple groups Ω2n+1(p) and PΩ4n(p) both occur as the Galois group of a Galois extension of the rationals for all integers n ≥ 2 and all primes p ≥ 5. We obtain our representations by studying families of twists of elliptic curves a...