Let H be a Hopf algebra. Ju and Cai introduced the notion of twisting of an H-module coalgebra. In this note, we study the relationship between twistings, crossed coproducts and Hopf-Galois coextensions. In particular, we show that a twisting of an H-Galois coextension remains H-Galois if the twisting is invertible.

The uniform distribution of the trace map lends itself very well to the construction of binary and non-binary codes from Galois fields and Galois rings. In this paper we study the distribution of the trace map with the argument ax over the Galois field GF (p, 2). We then use this distribution to construct two-weight, self-orthogonal, trace codes.

We prove Serre’s conjecture for the case of Galois representations of Serre’s weight 2 and level 1. We do this by combining the potential modularity results of Taylor and lowering the level for Hilbert modular forms with a Galois descent argument, properties of universal deformation rings, and the non-existence of p-adic Barsotti-Tate conductor 1 Galois representations proved in [Di3].

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.

This paper studies the structure of U(g)-Galois extensions. In particular, we use a result of Bell to construct a “PBW-like” free basis for faithfully flat U(g)-Galois extensions. We then move to non-faithfully flat extensions and propose a possible equivalent condition for a U(g)-extension to be Galois. We get a partial result for this.

We define comodule algebras and Galois extensions for actions of bialgebroids. Using just module conditions we characterize the Frobenius extensions that are Galois as depth two and right balanced extensions. As a corollary, we obtain characterizations of certain weak and ordinary Hopf-Galois extensions without reference to action in the hypothesis. 2000 AMS Subject Classification: 13B05, 16W30

We prove Serre’s conjecture for the case of Galois representations with Serre’s weight 2 and level 1. We do this by combining the potential modularity results of Taylor and lowering the level for Hilbert modular forms with a Galois descent argument, properties of universal deformation rings, and the non-existence of p-adic Barsotti-Tate conductor 1 Galois representations proved in [Di3].

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.

In this paper, we give new constructions of disjoint difference families from Galois rings. The constructions are based on choosing cosets of the unit group of a subring in the Galois ring GR(p2, p2s). Two infinite families of disjoint difference families are obtained from the Galois rings GR(p2, p4n) and GR(22, 22s).