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...

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...

Hopf–Galois extensions of rings generalize Galois extensions, with the coaction of a Hopf algebra replacing the action of a group. Galois extensions with respect to a group G are the Hopf–Galois extensions with respect to the dual of the group algebra of G . Rognes recently defined an analogous notion of Hopf–Galois extensions in the category of structured ring spectra, motivated by the fundame...

1. Throughout this paper, K will represent a division ring and L a galois division subring. We are interested in establishing a galois theory for the extension K/L when K/L is locally finite. In order to do this one must identify the galois subrings of K containing L. An example given by Jacobson [4] shows that not every such division subring is galois. However, we obtain that each subring subj...