We find a relationship between regular embeddings of G, an elementary abelian p-group of order p, into unipotent upper triangular matrices with entries in Fp and commutative dimension n degree 2 polynomial formal groups with nilpotent upper triangular structure matrices. We classify the latter when n = 3 up to linear isomorphism, and use that classification to determine the number of Hopf Galoi...

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...

Hopf Galois theory expands the classical Galois theory by considering the Galois property in terms of the action of the group algebra k[G] on K/k and then replacing it by the action of a Hopf algebra. We review the case of separable extensions where the Hopf Galois property admits a group-theoretical formulation suitable for counting and classifying, and also to perform explicit computations an...

We reduce certain proofs in [16, 11, 12] to depth two quasibases from one side only, a minimalistic approach which leads to a characterization of Galois extensions for finite projective bialgebroids without the Frobenius extension property. We prove that a proper algebra extension is a left T -Galois extension for some right finite projective left bialgebroid over some algebra R if and only if ...

In this paper, we investigate the properties of fuzzy Galois (dual Galois, residuated, and dual residuated) connections in a complete residuated lattice L. We give their examples. In particular, we study fuzzy Galois (dual Galois, residuated, dual residuated) connections induced by L-fuzzy relations.

Galois rings are regarded as “building blocks” of a finite commutative ring with identity. There have been many papers on classical error correction codes over Galois rings published. As an important warm-up before exploring quantum algorithms and quantum error correction codes over Galois rings, we study the quantum Fourier transform (QFT) over Galois rings and prove it can be efficiently pref...

Serre has asked if there are motives M with motivic Galois group of type G2 [Se3; pg 386]. This paper is the first step in a project to construct such a motive M , of rank 7 and weight 0, over the base field Q. Let G be the anisotropic form of G2 over Q, and let π = ⊗̂vπv be an automorphic representation of the adelic group G(A). At almost all primes p, the local representation πp is unramified ...

We classify the filtered modules with coefficients corresponding to two-dimensional potentially semi-stable p-adic representations of the Galois groups of p-adic fields under the assumptions that p is odd and the coefficients are large enough. Introduction Let p be an odd prime number, and let K be a p-adic field. The absolute Galois group of K is denoted by GK . By the fundamental theorem of C...

We prove a variety of results on the existence of automorphic Galois representations lifting a residual automorphic Galois representation. We prove a result on the structure of deformation rings of local Galois representations, and deduce from this and the method of Khare and Wintenberger a result on the existence of modular lifts of specified type for Galois representations corresponding to Hi...