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...

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...

For a long period the theory of modules over rings on the one hand and comodules and Hopf modules for coalgebras and bialgebras on the other side developed quite independently. In this talk we want to outline how ideas from module theory can be applied to enrich the theory of comodules and vice versa. For this we consider A-corings C with grouplike elements over a ring A, in particular Galois c...

A factorization of a Galois connection investigated earlier is used to give a definition of a connectedness-disconnectedness Galois connection that is free of the notion of constant morphism. A new notion of N -fixed morphism with respect to a classN of monomorphisms is presented. This is used to characterize the connectedness-disconnectedness Galois connection in the case that N is closed unde...

This thesis is concerned with the definition and the study of properties of homotopic Hopf-Galois extensions in the category Ch 0 k of chain complexes over a field k, equipped with its projective model structure. Given a differential graded k-Hopf algebra H of finite type, we define a homotopic H-Hopf-Galois extension to be a morphism ' : B ! A of augmented H-comodule dg-k-algebras, where B is ...

At present we do not know a general algorithm that will compute the Galois group of a linear differential equation with coefficients in a differential field k, even when k = Q̄(x), where Q̄ is the algebraic closure of the rational numbers. In contrast, algorithms for calculating the Galois group of a polynomial with coefficients in Q or Q̄(x) have been known for a long time (van der Waerden, 1953;...

For fields F of characteristic not p containing a primitive pth root of unity, we determine the Galois module structure of the group of pth-power classes of K for all cyclic extensions K/F of degree p. The foundation of the study of the maximal p-extensions of fields K containing a primitive pth root of unity is a group of the pth-power classes of the field: by Kummer theory this group describe...

We introduce (fuzzy) Galois connections with hedges. Fuzzy Galois connections are basic structures behind so-called formal concept analysis of data with fuzzy attributes. Introducing hedges to Galois connections means introducing two parameters. The parameters influence the size of the set of all the fixpoints of a Galois connection. In the sense of formal concept analysis, the fixpoints, calle...

In the present paper, we give necessary and sufficient conditions for a birational Galois section of a projective smooth curve over either the field of rational numbers or an imaginary quadratic field to be geometric. As a consequence, we prove that, over such a small number field, to prove the birational section conjecture for projective smooth curves, it suffices to verify that, roughly speak...

We describe some of the basic ideas of Galois theory for commutative S-algebras originally formulated by John Rognes. We restrict attention to the case of finite Galois groups. We describe the general framework developed by Rognes. Central rôles are played by the notion of strong duality and a trace or norm mapping constructed by Greenlees and May in the context of generalized Tate cohomology. ...