We say a tame Galois field extension L/K with Galois group G has trivial Galois module structure if the rings of integers have the property that OL is a free OK [G]-module. The work of Greither, Replogle, Rubin, and Srivastav shows that for each algebraic number field other than the rational numbers there will exist infinitely many primes l so that for each there is a tame Galois field extensio...

As Lie group theory has developed it has also become more and more pervasive in its influence on other mathematical disciplines. The original founder of this theory was a Norwegian, Marius Sophus Lie, who was born in Nordfjordeid, 1842. In order to understand the background and motivation to Lie's work we must go further back. A central problem in algebra at the end of the 18th century was that...

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...

We investigate the Galois coverings of weakly shod algebras. For a weakly shod algebra not quasi-tilted of canonical type, we establish a correspondence between its Galois coverings and the Galois coverings of its connecting component. As a consequence we show that a weakly shod algebra which is not quasi-tilted of canonical type is simply connected if and only if its first Hochschild cohomolog...

Serre’s conjecture relates two-dimensional odd irreducible Galois representations over F̄p to modular forms. We discuss a generalization of this conjecture to higher-dimensional Galois representations. In particular, for n-dimensional Galois representations which are irreducible when restricted to the decomposition group at p, we strengthen a conjecture of Ash, Doud, and Pollack. We then give co...

In this note we introduce the notion of a group action on a set and use it to prove a number of theorems about p-groups and the Sylow theorems. For undefined terms see any decent book on group theory. The theory of p-groups and the Sylow theorems have a number of applications in Galoistheory. The nice structure of p-groups will translate via the fundamental theorem of Galois theory to nice stru...

Past studies of the Brauer group of a scheme tells us the importance of the interrelationship among Brauer groups of its finite étale coverings. In this paper, we consider these groups simultaneously, and construct an integrated object “BrauerMackey functor”. We realize this as a cohomological Mackey functor on the Galois category of finite étale coverings. For any finite étale covering of sche...

We show that the Galois group Gal(Q̄/Q) operates faithfully on the set of connected components of the moduli spaces of surfaces of general type, and also that for each element σ ∈ Gal(Q̄/Q) different from the identity and from complex conjugation, there is a surface of general type such that X and the Galois conjugate variety Xσ have nonisomorphic fundamental groups. The result was announced by t...

In this article new cases of the inverse Galois problem are established. The main result is that for a fixed integer n, there is a positive density set of primes p such that PSL2(Fpn ) occurs as the Galois group of some finite extension of the rational numbers. These groups are obtained as projective images of residual modular Galois representations. Moreover, families of modular forms are cons...

The non-Abelian finite simple groups and their automorphism groups play a crucial role in an inductive approach to the inverse problem of Galois theory. The rigidity method (see, for example, Malle and Matzat, 1999) has proved very efficient for deducing the existence of Galois extensions with such groups, as well as for the construction of polynomials generating such extensions. Nevertheless, ...