We show that the simple group PSL2(Fp) occurs as the Galois group of an extension of the rationals for all primes p ≥ 5. We obtain our Galois extensions by studying the Galois action on the second étale cohomology groups of a specific elliptic surface.

In this paper we present a new method for determining the Galois group of a square free univariate polynomial. This method makes use of a priori computation of the Galois group of the factors of its resolvents, and can also be used for the Galois inverse problem.

l-adic Étale Theory Hodge Theory Category of l-adic Abelian category MHR Galois modules of real mixed Hodge strucrures Galois group Hodge Galois group GHod := Gal(Q/Q) Galois group of the category MHR Gal(Q/Q) acts on H∗ et(X,Ql), H∗(X(C),R) has a functorial where X is a variety over Q real mixed Hodge structure étale site ?? Gal(Q/Q) acts on the étale site, and thus ?? on categories of étale s...

Introduction. Galois extensions of noncommutative rings were introduced in 1964 by Teruo Kanzaki [13]. These algebraic objects generalize to noncommutative rings the classical Galois extensions of fields and the Galois extensions of commutative rings due to Auslander and Goldman [1]. At the same time they also turn out to be fundamental examples of Hopf-Galois extensions; these were first consi...

The classical Galois theory of fields and the classification of covering spaces of a path-connected, locally path-connected, and semi-locally simply connected space (which will be referred to as the Galois theory of covering spaces) appear very similar. We study the connection of these two Galois theories by generalizing them in categorical language as equivalences of certain categories. This i...

Consider any nilpotent group G of finite odd order. We ask if we can always find a galois extension K of Q such that Gal(K/Q) ∼= G. This is the famous Inverse Galois Problem applied to nilpotent groups of finite odd order. By solving the Group Extension Problem and the Embedding Problem, two problems that are related to the Inverse Galois Problem, we show that such a K always exists. A major re...

We prove that any Galois extension of commutative rings with normal basis and abelian Galois group of odd order has a self dual normal basis. Also we show that if S/R is an unramified extension of number rings with Galois group of odd order and R is totally real then the normal basis does not exist for S/R.

We study the middle convolution of local systems on the punctured affine line in the singular and the étale case. We give a motivic interpretation of the middle convolution which yields information on the occurring determinants. Finally, we use these methods to realize special linear groups regularly as Galois groups over Q(t). Introduction If K is a field, then we set GK := Gal(K/K), where K d...

James Ax showed that, in each characteristic, there is a natural bijection from the space of complete theories of pseudo-finite fields, in first order logic, to the set of conjugacy classes of procyclic subgroups of the absolute Galois group of the prime field. I show that when the set of subgroups of a profinite group is considered to have the Vietoris (a.k.a. hyperspace, finite, exponential, ...

let $cr_{n}(f)$ denote the algebra of $n times n$ circulant matrices over the field $f$. in this paper, we study the unit group of $cr_{n}(f_{p^m})$, where $f_{p^m}$ denotes the galois field of order $p^{m}$, $p$ prime.