Let S be the left R-bialgebroid of a depth two extension with cen-tralizer R. We show that the left endomorphism ring of depth two extension, not necessarily balanced, is a left S-Galois extension of A op. Looking to examples of depth two, we establish that a Hopf subalgebra is normal if and only if it is a Hopf-Galois extension. We find a class of examples of the alternative Hopf algebroids in...

RÉSUMÉ – Organisation de données probabilistes par des treillis de Galois Dans cet article, nous nous intéressons à l’organisation de données probabilistes par des treillis de Galois. Deux correspondances de Galois sont établies, en définissant de façon appropriée les opérateurs de généralisation et d'extension. Ces correspondances permettent de construire deux treillis, appelés treillis de l’u...

Let A and B be R-algebras with automorphism groups G and H respectively. Denote the order of G by n and the order of H by m for some integers n and m. Assume n and m are invertible in R. Then, A⊗R B is a Galois R-algebra with Galois group G × H if and only if A and B are Galois R-algebras with Galois groups G and H respectively. Thus an equivalent condition for a central Galois algebra in terms...

We construct explicitly APF extensions of finite extensions of Qp for which the Galois group is not a p-adic Lie group and which do not have any open subgroup with Zp-quotient. Let K be a finite extension of Qp and L/K be a Galois totally ramified pro-pextension. If the Galois group of L/K is a p-adic Lie group it is known from Sen ([Sen]) that the sequence of upper ramification breaks of L/K i...

The concept of Galois connection between power sets is generalized from the point of view of fuzzy logic. Studied is the case where the structure of truth values forms a complete residuated lattice. It is proved that fuzzy Galois connections are in one-to-one correspondence with binary fuzzy relations. A representation of fuzzy Galois connections by (classical) Galois connections is provided. 1...

We prove a direct image theorem stating that the direct image of a Galois formula by a morphism of difference schemes is equivalent to a Galois formula modulo the theory ACFA of existentially closed difference fields. As a consequence, we obtain an effective quantifier elimination procedure and a precise algebraic-geometric description of definable sets over existentially closed difference fiel...

Coalgebra-Galois extensions generalise Hopf-Galois extensions, which can be viewed as non-commutative torsors. In this paper it is analysed when a coalgebra-Galois extension is a separable, split, or strongly separable extension.

Galois is a domain specific language supported by the Galculator interactive proof-assistant prototype. Galculator uses an equational approach based on Galois connections with indirect equality as an additional inference rule. Galois allows for the specification of different theories in a point-free style by using fork algebras, an extension of relation algebras with expressive power of first-o...

4 Introduction to Galois Theory 2 4.1 Polynomial Rings . . . . . . . . . . . . . . . . . . . . . . . . . 2 4.2 Gauss’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . 5 4.3 Eisenstein’s Irreducibility Criterion . . . . . . . . . . . . . . . 6 4.4 Field Extensions and the Tower Law . . . . . . . . . . . . . . 6 4.5 Algebraic Field Extensions . . . . . . . . . . . . . . . . . . . . 8 4....

We introduce a concept and develop a theory of Galois subalgebras in skew semigroup rings. Proposed approach has a strong impact on the representation theory, first of all the theory of Harish-Chandra modules, of many infinite dimensional algebras including the Generalized Weyl algebras, the universal enveloping algebras of reductive Lie algebras, their quantizations, Yangians etc. In particula...