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A brief heuristic explanation is given of recent work with Juergen Fuchs, Beatriz Gato-Rivera and Christoph Schweigert on the construction of modular invariant partition functions from Galois symmetry in conformal field theory. A generalization, which we call quasi-Galois symmetry, is also described. As an application of the latter, the invariants of the exceptional algebras at level $g$ (for example $E_8$ level 30) expected from conformal embeddings are presented. [Contribution to the Proceedings of the International Symposium on the Theory of Elementary Particles Wendisch-Rietz, August 30 - September 3, 1994]

We reduce certain proofs in math.RA/0108067, math.RA/0408155, and math.QA/0409589 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 it is a left depth two and left balanced extension. Exchanging left and right in this statement, we have a characterization of right Galois extensions for left finite projective right bialgebroids. 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 characterize finite weak Hopf-Galois extensions using an alternate Galois canonical mapping with several corollaries: that these are depth two and that surjectivity of the Galois mapping implies its bijectivity.

We reduce certain proofs in math.RA/0108067, math.RA/0408155, and math.QA/0409589 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 it is a left depth two and left balanced extension. Exchanging left and right in this statement, we have a characterization of right Galois extensions for left finite projective right bialgebroids. 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 characterize finite weak Hopf-Galois extensions using an alternate Galois canonical mapping with several corollaries: that these are depth two and that surjectivity of the Galois mapping implies its bijectivity.

We reduce certain proofs in math.RA/0108067, math.RA/0408155, and math.QA/0409589 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 it is a left depth two and left balanced extension. Exchanging left and right in this statement, we have a characterization of right Galois extensions for left finite projective right bialgebroids. 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 characterize finite weak Hopf-Galois extensions using an alternate Galois canonical mapping with several corollaries: that these are depth two and that surjectivity of the Galois mapping implies its bijectivity.

We reduce certain proofs in math.RA/0108067, math.RA/0408155, and math.QA/0409589 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 it is a left depth two and left balanced extension. Exchanging left and right in this statement, we have a characterization of right Galois extensions for left finite projective right bialgebroids. 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 characterize finite weak Hopf-Galois extensions using an alternate Galois canonical mapping with several corollaries: that these are depth two and that surjectivity of the Galois mapping implies its bijectivity.

We prove that two arithmetically significant extensions of a field F coincide if and only if the Witt ring WF is a group ring Z/n[G]. Furthermore, working modulo squares with Galois groups which are 2-groups, we establish a theorem analogous to Hilbert's Theorem 90 and show that an identity linking the cohomological dimension of the Galois group of the quadratic closure of F, the length of a filtration on a certain module over a Galois group, and the dimension over Z/2 of the square class group of the field holds for a number of interesting families of fields. Finally we discuss the cohomology of a particular Galois group in a topological context.

In this paper we introduce a notion of integrability in the non autonomous sense. For the cases of 1 + 1/2 degrees of freedom and quadratic homogeneous Hamiltonians of 2 + 1/2 degrees of freedom we prove that this notion is equivalent to the classical complete integrability of the system in the extended phase space. For the case of quadratic homogeneous Hamiltonians of 2 + 1/2 degrees of freedom we also give a reciprocal of the Morales-Ramis result. We classify those systems by terms of symplectic change of frames involving algebraic functions of time, and give their canonical forms.

We study parameterized linear differential equations with coefficients depending meromorphically upon the parameters. As a main result, analogous to Ramis's theorem in the unparameterized case, we show that the parameterized monodromy, the parameterized exponential torus and the parameterized Stokes operators are topological generators of the parameterized differential Galois group in Kolchin's topology. We prove an analogous result for the global parameterized differential Galois group and we give a sufficient condition on a group for being a global parameterized differential Galois group. As an application, we give a characterization of the completely integrable equations and we give a partial answer to a question of Sibuya. Moreover, we prove a parameterized Turrittin theorem, which allow to show directly that the Galois group descends to a smaller field whose field of constant is only algebraically closed rather than differentially closed.

We elaborate on the representation theorems of topoi as topoi of discrete actions of various kinds of localic groups and groupoids. We introduce the concept of "proessential point" and use it to give a new characterization of pointed Galois topoi. We establish a hierarchy of connected topoi: [1. essentially pointed Atomic = locally simply connected], [2. proessentially pointed Atomic = pointed Galois], [3. pointed Atomic]. These topoi are the classifying topos of, respectively: 1. discrete groups, 2. prodiscrete localic groups, and 3. general localic groups. We analyze also the unpoited version, and show that for a Galois topos, may be pointless, the corresponding groupoid can also be considered, in a sense, the groupoid of "points". In the unpointed theories, these topoi classify, respectively: 1. connected discrete groupoids, 2. connected (may be pointless) prodiscrete localic groupoids, and 3. connected groupoids with discrete space of objects and general localic spaces of hom-sets, when the topos has points (we do not know the class of localic groupoids that correspond to pointless connected atomic topoi). We comment and develop on Grothendieck's galois theory and its generalization by Joyal-Tierney, and work by other authors on these theories.

We elaborate on the representation theorems of topoi as topoi of discrete actions of various kinds of localic groups and groupoids. We introduce the concept of "proessential point" and use it to give a new characterization of pointed Galois topoi. We establish a hierarchy of connected topoi: [1. essentially pointed Atomic = locally simply connected], [2. proessentially pointed Atomic = pointed Galois], [3. pointed Atomic]. These topoi are the classifying topos of, respectively: 1. discrete groups, 2. prodiscrete localic groups, and 3. general localic groups. We analyze also the unpoited version, and show that for a Galois topos, may be pointless, the corresponding groupoid can also be considered, in a sense, the groupoid of "points". In the unpointed theories, these topoi classify, respectively: 1. connected discrete groupoids, 2. connected (may be pointless) prodiscrete localic groupoids, and 3. connected groupoids with discrete space of objects and general localic spaces of hom-sets, when the topos has points (we do not know the class of localic groupoids that correspond to pointless connected atomic topoi). We comment and develop on Grothendieck's galois theory and its generalization by Joyal-Tierney, and work by other authors on these theories.

For a finite Galois extension of fields L/k with Galois group G, we study a functor from the G-equivariant stable homotopy category to the stable motivic homotopy category over k induced by the classical Galois correspondence. We show that after completing at a prime and eta (the motivic Hopf map) this results in a full and faithful embedding whenever k is real closed and L = k[i]. It is a full and faithful embedding after eta-completion if a motivic version of Serre's finiteness theorem is valid. We produce strong necessary conditions on the field extension L/k for this functor to be full and faithful. Along the way, we produce several results on the stable C_2-equivariant Betti realization functor and prove convergence theorems for the p-primary C_2-equivariant Adams spectral sequence.

We show that there exists a Galois correspondence between subalgebras of an H-comodule algebra A over a base ring R and generalised quotients of a Hopf algebra H. We also show that Q-Galois subextensions are closed elements of the constructed Galois connection. Then we consider the theory of coextensions of H-module coalgebras. We construct Galois theory for them and we prove that H-Galois coextensions are closed. We apply the obtained results to the Hopf algebra itself and we show a simple proof that there is a bijection correspondence between right ideal coideals of H and its left coideal subalgebras when H is finite dimensional. Furthermore we formulate necessary and sufficient conditions when the Galois correspondence is a bijection for arbitrary Hopf algebras. We also present new conditions for closedness of subalgebras and generalised quotients when A is a crossed product.

We consider a quantum theory based on a Galois field. In this approach infinities cannot exist, the cosmological constant problem does not arise, and one irreducible representation (IR) of the symmetry algebra splits into independent IRs describing a particle an its antiparticle only in the approximation when de Sitter energies are much less than the characteristic of the field. As a consequence, the very notions of particles and antiparticles are only approximate and such additive quantum numbers as the electric, baryon and lepton charges are conserved only in this approximation. There can be no neutral elementary particles and the spin-statistics theorem can be treated simply as a requirement that standard quantum theory should be based on complex numbers.

In previous papers, we defined $q$-analogues of alien derivations for linear analytic $q$-difference equations with integral slopes and proved a density theorem (in the Galois group) and a freeness theorem. In this paper, we completely describe the wild fundamental group and apply this result to the inverse problem in $q$-difference Galois theory. The new version contains an appendix on pronilpotent completion and the main result on the direct problem is made more precise. (Submitted for publication)

We present a Galois theory connecting finitary operations with pairs of finitary relations one of which is contained in the other. The Galois closed sets on both sides are characterised as locally closed subuniverses of the full iterative function algebra (semiclones) and relation pair clones, respectively. Moreover, we describe the modified closure operators if only functions and relation pairs of a certain bounded arity, respectively, are considered.

Let $K$ be a finite extension of $\mathbb{Q}_p$, and choose a uniformizer $\pi\in K$, and put $K_\infty:=K(\sqrt[p^\infty]{\pi})$. We introduce a new technique using restriction to $\Gal(\ol K/K_\infty)$ to study flat deformation rings. We show the existence of deformation rings for $\Gal(\ol K/K_\infty)$-representations ``of height $\leqslant h$'' for any positive integer $h$, and we use them to give a variant of Kisin's proof of connected component analysis of a certain flat deformation rings, which was used to prove Kisin's modularity lifting theorem for potentially Barsotti-Tate representations. Our proof does not use the classification of finite flat group schemes, so it avoids Zink's theory of windows and displays when $p=2$. This $\Gal(\ol K/K_\infty)$-deformation theory has a good analogue in positive characteristics analogue of crystalline representations in the sense of Genestier-Lafforgue. In particular, we obtain a positive characteristic analogue of crystalline deformation rings, and can analyze their local structure.

For a number field K with absolute Galois group G_K, we consider the action of G_K on the infinite tree of preimages of a point in K under a degree-two rational function phi, with particular attention to the case when phi commutes with a non-trivial Mobius transfomation. In a sense this is a dynamical systems analogue to the l-adic Galois representation attached to an elliptic curve, with particular attention to the CM case. Using a result about the discriminants of numerators of iterates of phi, we give a criterion for the image of the action to be as large as possible. This criterion is in terms of the arithmetic of the forward orbits of the two critical points of phi. In the case where phi commutes with a non-trivial Mobius transfomation, there is in effect only one critical orbit, and we give a modified version of our maximality criterion. We prove a Serre-type finite-index result in many cases of this latter setting.

This is an essay to accompany the author's lecture at the introductory workshop on `Nonabelian fundamental groups in arithmetic geometry' at the Newton Institute, Cambridge in July, 2009.

We develop a Galois (descent) theory for comonads within the framework of bicategories. We give generalizations of Beck's theorem and the Joyal-Tierney theorem. Many examples are provided, including classical descent theory, Hopf-Galois theory over Hopf algebras and Hopf algebroids, Galois theory for corings and group-corings, and Morita-Takeuchi theory for corings. As an application we construct a new type of comatrix corings based on (dual) quasi bialgebras.

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 and explicit descriptions of all the ingredients involved in a Hopf Galois structure. At the end we give just a glimpse of how this theory is used in the context of Galois module theory for wildly ramified extensions.

We investigate the relations in Galois groups of maximal p-extensions of fields, the structure of their natural filtrations, and their relationship with the Bloch-Kato conjecture proved by Rost and Voevodsky with Weibel's patch. Our main focus is on the third degree, but we provide examples for all degrees.

We combine the projective geometry approach to Schroedinger equations on the circle and differential Galois theory with the theory of Poisson Lie groups to construct a natural Poisson structure on the space of wave functions (at the zero energy level). Applications to KdV-like nonlinear equations are discussed. The same approach is applied to second order difference operators on a one-dimensional lattice, yielding an extension of the lattice Poisson Virasoro algebra.

The purpose of this paper is to connect two subjects: the theory of quantum integrable systems (complete commutative rings of differential operators), and differential Galois theory. We define quantum completely integrable systems (QCIS), algebraically integrable QCIS, the differential Galois group of a QCIS. We show that the differential Galois group is always reductive and that a QCIS is algebraically integrable if and only if its differential Galois group is commutative. In particular, we show that a differential operator L in one variable is algebraic in the sense of Krichever (i.e. finite-zone) if and only if the differential Galois group of the differential equation Lf=af is commutative for a generic number a. As a by-product, we obtain a proof of the Veselov-Chalyh conjecture on the algebraic integrability of the elliptic Calogero-Moser system.

The purpose of this paper is to connect two subjects: the theory of quantum integrable systems (complete commutative rings of differential operators), and differential Galois theory. We define quantum completely integrable systems (QCIS), algebraically integrable QCIS, the differential Galois group of a QCIS. We show that the differential Galois group is always reductive and that a QCIS is algebraically integrable if and only if its differential Galois group is commutative. In particular, we show that a differential operator L in one variable is algebraic in the sense of Krichever (i.e. finite-zone) if and only if the differential Galois group of the differential equation Lf=af is commutative for a generic number a. As a by-product, we obtain a proof of the Veselov-Chalyh conjecture on the algebraic integrability of the elliptic Calogero-Moser system.

The purpose of this paper is to connect two subjects: the theory of quantum integrable systems (complete commutative rings of differential operators), and differential Galois theory. We define quantum completely integrable systems (QCIS), algebraically integrable QCIS, the differential Galois group of a QCIS. We show that the differential Galois group is always reductive and that a QCIS is algebraically integrable if and only if its differential Galois group is commutative. In particular, we show that a differential operator L in one variable is algebraic in the sense of Krichever (i.e. finite-zone) if and only if the differential Galois group of the differential equation Lf=af is commutative for a generic number a. As a by-product, we obtain a proof of the Veselov-Chalyh conjecture on the algebraic integrability of the elliptic Calogero-Moser system.

The purpose of this paper is to connect two subjects: the theory of quantum integrable systems (complete commutative rings of differential operators), and differential Galois theory. We define quantum completely integrable systems (QCIS), algebraically integrable QCIS, the differential Galois group of a QCIS. We show that the differential Galois group is always reductive and that a QCIS is algebraically integrable if and only if its differential Galois group is commutative. In particular, we show that a differential operator L in one variable is algebraic in the sense of Krichever (i.e. finite-zone) if and only if the differential Galois group of the differential equation Lf=af is commutative for a generic number a. As a by-product, we obtain a proof of the Veselov-Chalyh conjecture on the algebraic integrability of the elliptic Calogero-Moser system.

This note is a development of our two previous papers, arXiv:1212.3392v1 and 1306.3660v1. The fundamental question is whether there exists a Galois theory, in which the Galois group is a quantum group. For a linear equations with respect to a Hopf algebra, we arrived at a final form if the base field consists of constants. In this case, we have non-commutative Picard-Vessiot rings and asymmetric Tannaka theory. For non-linear equations there are examples that might make us optimistic.

We establish a precise relation between Galois theory in its motivic form with the mathematical theory of perturbative renormalization (in the minimal subtraction scheme with dimensional regularization). We identify, through a Riemann-Hilbert correspondence based on the Birkhoff decomposition and the t'Hooft relations, a universal symmetry group (the "cosmic Galois group" suggested by Cartier), which contains the renormalization group and acts on the set of physical theories. This group is closely related to motivic Galois theory. We construct a universal singular frame of geometric nature, in which all divergences disappear. The paper includes a detailed overview of the work of Connes-Kreimer and background material on the main quantum field theoretic and algebro-geometric notions involved. We give a complete account of our results announced in math.NT/0409306.

Galois theory is a central part of algebra, dealing with symmetries between solutions of algebraic equations in one variable. This collection of papers brings together articles from some of the world's leading experts in this field. Topics center around the Inverse Galois Problem, comprising the full range of methods and approaches in this area, making this an invaluable resource for all those whose research involves Galois theory. Title: Aspects of Galois Theory Author(s): Helmut Voelklein (editor), J. G. Thompson (editor), David Harbater (editor), Peter Müller (editor) Series: London Mathematical Society Lecture Note Series 256 Language: English Year: 1999 Publisher: Cambridge University Press ISBN: 0521637473, 9780521637473 Pages: 292 (bibliographic) / 289 (file) Format: PDF File size: 11.40 Mb (11951047 bytes) Time added / time modified: 2023-04-16 09:21:29 Commentary: OCR'd with ABBYY Finereader (not proofread)

We explore connections between birational anabelian geometry and abstract projective geometry. One of the applications is a proof of a version of the birational section conjecture.

In this paper we develop a differential Galois theory for algebraic Lie-Vessiot systems in algebraic homogeneous spaces. Lie-Vessiot systems are non autonomous vector fields that are linear combinations with time-dependent coefficients of fundamental vector fields of an algebraic Lie group action. Those systems are the building blocks for differential equations that admit superposition of solutions. Lie-Vessiot systems in algebraic homogeneous spaces include the case of linear differential equations. Therefore, the differential Galois theory for Lie-Vessiot systems is an extension of the classical Picard-Vessiot theory. In particular, algebraic Lie-Vessiot systems are solvable in terms of Kolchin's strongly normal extensions. Therefore, strongly normal extensions are geometrically interpreted as the fields of functions on principal homogeneous spaces over the Galois group. Finally we consider the problem of integrability and solvability of automorphic differential equations. Our main tool is a classical method of reduction, somewhere cited as Lie reduction. We develop and algebraic version of this method, that we call Lie-Kolchin reduction. Obstructions to the application are related to Galois cohomology.

We generalize a work of Iovita-Zaharescu on the Galois theory of B_dR^+ to the imperfect residue field case. The proof is based on a structure theorem of Colmez's higher Kahler differentials.

This is a concise survey of links between Galois module theory and class field theory (CFT). It explores various uses of CFT in Galois module theory, it comments on the absence of CFT in contexts where it might be expected to play a role and indicates some lines of research that might bring CFT to play a more prominent role in Galois module theory.

We outline the proof of a theorem of M. Baker and A. Tamagawa that gives a complete description of the torsion points on a modular curve embedded in its Jacobian using the notion of an `almost rational torsion point.'

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.

This video is an introduction to Galois Theory, which spells out a beautiful correspondence between fields and their symmetry groups. __ SOURCES and REFERENCES for Further Reading! This video is a quick-and-dirty introduction to Galois theory. But as with any quick introduction, there are details that I gloss over for the sake of brevity. To learn these details rigorously, I've listed a few resources down below. (a) Galois Theory Galois Theory notes by Tom Leinster: These notes are by far the best resource out there for learning the subject. They’re completely rigorous, but they’re also written in a very reader-friendly way with lots of examples and motivation. (See link here: https://www.maths.ed.ac.uk/~tl/gt/gt.pdf) This playlist on Galois Theory by Professor Richard Borcherds is a gem. It explains Galois Theory from the ground up, rigorously, in almost complete generality. (https://www.youtube.com/playlist?list=PL8yHsr3EFj53Zxu3iRGMYL_89GDMvdkgt) (b) Group Theory Group Theory lectures: This playlist by Professor Benedict Gross is a beauty. It goes through the entire group theory syllabus from the ground up, and Professor Gross is a masterful lecturer. (see link here: https://www.youtube.com/playlist?list=PLelIK3uylPMGzHBuR3hLMHrYfMqWWsmx5) MUSIC CREDITS: The song is called “Taking Flight”, by Vince Rubinetti. https://www.vincentrubinetti.com/ THANK YOUs: Extra special thanks to Davide Radaelli and Grant Sanderson for helpful conversations while making this video! SOFTWARE USED: Adobe Premiere Pro for Editing Follow me! Twitter: @00aleph00 Instagram: @00aleph00 Intro: (0:00) What is the square root of 2?: (1:08) Fields and Automorphisms: (6:04) Examples: (8:55) Group Theory: (16:34) The Fundamental Theorem: (18:25)

This video is an introduction to Galois Theory, which spells out a beautiful correspondence between fields and their symmetry groups. __ SOURCES and REFERENCES for Further Reading! This video is a quick-and-dirty introduction to Galois theory. But as with any quick introduction, there are details that I gloss over for the sake of brevity. To learn these details rigorously, I've listed a few resources down below. (a) Galois Theory Galois Theory notes by Tom Leinster: These notes are by far the best resource out there for learning the subject. They’re completely rigorous, but they’re also written in a very reader-friendly way with lots of examples and motivation. (See link here: https://www.maths.ed.ac.uk/~tl/gt/gt.pdf) This playlist on Galois Theory by Professor Richard Borcherds is a gem. It explains Galois Theory from the ground up, rigorously, in almost complete generality. (https://www.youtube.com/playlist?list=PL8yHsr3EFj53Zxu3iRGMYL_89GDMvdkgt) (b) Group Theory Group Theory lectures: This playlist by Professor Benedict Gross is a beauty. It goes through the entire group theory syllabus from the ground up, and Professor Gross is a masterful lecturer. (see link here: https://www.youtube.com/playlist?list=PLelIK3uylPMGzHBuR3hLMHrYfMqWWsmx5) MUSIC CREDITS: The song is called “Taking Flight”, by Vince Rubinetti. https://www.vincentrubinetti.com/ THANK YOUs: Extra special thanks to Davide Radaelli and Grant Sanderson for helpful conversations while making this video! SOFTWARE USED: Adobe Premiere Pro for Editing Follow me! Twitter: @00aleph00 Instagram: @00aleph00 Intro: (0:00) What is the square root of 2?: (1:08) Fields and Automorphisms: (6:04) Examples: (8:55) Group Theory: (16:34) The Fundamental Theorem: (18:25)

This video is an introduction to Galois Theory, which spells out a beautiful correspondence between fields and their symmetry groups. __ SOURCES and REFERENCES for Further Reading! This video is a quick-and-dirty introduction to Galois theory. But as with any quick introduction, there are details that I gloss over for the sake of brevity. To learn these details rigorously, I've listed a few resources down below. (a) Galois Theory Galois Theory notes by Tom Leinster: These notes are by far the best resource out there for learning the subject. They’re completely rigorous, but they’re also written in a very reader-friendly way with lots of examples and motivation. (See link here: https://www.maths.ed.ac.uk/~tl/gt/gt.pdf) This playlist on Galois Theory by Professor Richard Borcherds is a gem. It explains Galois Theory from the ground up, rigorously, in almost complete generality. (https://www.youtube.com/playlist?list=PL8yHsr3EFj53Zxu3iRGMYL_89GDMvdkgt) (b) Group Theory Group Theory lectures: This playlist by Professor Benedict Gross is a beauty. It goes through the entire group theory syllabus from the ground up, and Professor Gross is a masterful lecturer. (see link here: https://www.youtube.com/playlist?list=PLelIK3uylPMGzHBuR3hLMHrYfMqWWsmx5) MUSIC CREDITS: The song is called “Taking Flight”, by Vince Rubinetti. https://www.vincentrubinetti.com/ THANK YOUs: Extra special thanks to Davide Radaelli and Grant Sanderson for helpful conversations while making this video! SOFTWARE USED: Adobe Premiere Pro for Editing Follow me! Twitter: @00aleph00 Instagram: @00aleph00 Intro: (0:00) What is the square root of 2?: (1:08) Fields and Automorphisms: (6:04) Examples: (8:55) Group Theory: (16:34) The Fundamental Theorem: (18:25)

galois-theory-and-applications

ABSTRACT: Suppose every intrinsic, trivially additive topos is closed. It is well known that H = -INFINITY. We show that is invertible. It is not yet known whether q' is part of |jSIGMA|, although [20] does address the issue of injectivity. Recent developments in absolute calculus [20] have raised the question of whether v(u) != 1.

Let us take for granted that L_{K(n)}S^0 --> E_n is some kind of a G_n-Galois extension. Of course, this is in the setting of continuous G_n-spectra. How much structure does this continuous G-Galois extension have? How much structure does one want to build into this notion to obtain useful conclusions? If the author's conjecture that ``E_n/I, for a cofinal collection of I's, is a discrete G_n-symmetric ring spectrum" is true, what additional structure does this give the continuous G_n-Galois extension? Is it useful or merely beautiful? This paper is an exploration of how to answer these questions. This preprint arose as a letter to John Rognes, whom he thanks for a helpful conversation in Rosendal. This paper was written before John's preprints (the initial version and the final one) on Galois extensions were available.

Given a braided tensor *-category C with conjugate (dual) objects and irreducible unit together with a full symmetric subcategory S we define a crossed product C\rtimes S. This construction yields a tensor *-category with conjugates and an irreducible unit. (A *-category is a category enriched over Vect_C with positive *-operation.) A Galois correspondence is established between intermediate categories sitting between C and C\rtimes S and closed subgroups of the Galois group Gal(C\rtimes S/C)=Aut_C(C\rtimes S) of C, the latter being isomorphic to the compact group associated to S by the duality theorem of Doplicher and Roberts. Denoting by D\subset C the full subcategory of degenerate objects, i.e. objects which have trivial monodromy with all objects of C, the braiding of C extends to a braiding of C\rtimes S iff S\subset D. Under this condition C\rtimes S has no degenerate objects iff S=D. If the original category C is rational (i.e. has only finitely many equivalence classes of irreducible objects) then the same holds for the new one. The category C\rtimes D is called the modular closure of C since in the rational case it is modular, i.e. gives rise to a unitary representation of the modular group SL(2,Z). (In passing we prove that every braided tensor *-category with conjugates automatically is a ribbon category, i.e. has a twist.) If all simple objects of S have dimension one the structure of the category C\rtimes S can be clarified quite explicitly in terms of group cohomology.

We prove the Galois correspondence between the subgroups of a finite automorphism group G of a simple vertex operator algebra V and the vertex operator subalgebras of V containing the set V^G of G-invariants.

The main aim of this course is to study groups of automorphisms of �eld extensions, their �xed sub�elds, and the relationship between the structure of �eld extensions and the structure of the associated groups. We will pay special attention to the question whether a general polynomial has roots expressable by a formula consisting of �eld operations and taking nth roots, as in formula (1) for the quadratic polynomial. It was known by the 16th century that such a formula exists if deg f � 4(Tartaglia, Cardano, Ferrari). In contrast, it was discovered in the 19th century that there is usually no such formula for deg f � 5 (Ru_ni, Abel, Galois). We will prove these results as an application of our structure theory of �eld extensions

The main aim of this course is to study groups of automorphisms of �eld extensions, their �xed sub�elds, and the relationship between the structure of �eld extensions and the structure of the associated groups. We will pay special attention to the question whether a general polynomial has roots expressable by a formula consisting of �eld operations and taking nth roots, as in formula (1) for the quadratic polynomial. It was known by the 16th century that such a formula exists if deg f � 4(Tartaglia, Cardano, Ferrari). In contrast, it was discovered in the 19th century that there is usually no such formula for deg f � 5 (Ru_ni, Abel, Galois). We will prove these results as an application of our structure theory of �eld extensions

Includes bibliographical references (page 245)

Includes bibliographical references (page 245)

Includes bibliographical references (page 245)

We introduce the concept of homotopy equivalence for Hopf Galois extensions and make a systematic study of it. As an application we determine all H-Galois extensions up to homotopy equivalence in the case when H is a Drinfeld-Jimbo quantum group.

1Letters of a Woman Homesteader

By Elinore Pruitt Stewart

The writer of the following letters is a young woman who lost her husband in a railroad accident and went to Denver to seek support for herself and her two-year-old daughter, Jerrine. Turning her hand to the nearest work, she went out by the day as house-cleaner and laundress. Later, seeking to better herself, she accepted employment as a housekeeper for a well-to-do Scotch cattle-man, Mr. Stewart, who had taken up a quarter-section in Wyoming. The letters, written through several years to a former employer in Denver, tell the story of her new life in the new country. They are genuine letters, and are printed as written, except for occasional omissions and the alteration of some of the names. (Publishers’ Note, May 1914)

2Perfect Behavior

By Donald Ogden Stewart

A humorous guide to manners and etiquette for ladies and gentlemen in a social "crises," published in 1922. (Introduction by Samanem)

Maria W. Stewart was America's first black woman political writer. Between 1831 and 1833, she gave four speeches on the topics of slavery and women's rights. <em>Meditations From The Pen of Mrs. Maria W. Stewart</em>—published in 1879, shortly before her death—is a collection of those speeches as well as her memoir, some meditations and prayers. They are political, poetical and sermon all at the same time; but in the milieu in which she lectured, they were a critically important part of the abolitionist movement years before the contributions of others such as Frederick Douglass and Sojourner Truth. Her speeches and essays espoused a return to Christian values and morality, but also proposed fundamental changes in gender roles in the midst of tremendous public opposition to the rights of blacks and of women. (Introduction by James K. White)

4Uncle Josh's Punkin Centre Stories

By Cal Stewart

A collection of comedic short stories from the perspective of an old country man. (Summary by Philip Martin)

5Letters on an Elk Hunt

By Elinore Pruitt Stewart

This is a sequel to Letters of a Woman Homesteader in which Elinore Rupert (Pruitt) Stewart describes her arrival and early years on a Burntfork Wyoming ranch in 1909-1913. The letters are written to her elderly friend, Mrs. Coney, in Denver. In the present collection of letters, Elinore describes a lively excursion on horseback and wagon into the Wyoming wilderness during July-October 1914. Her traveling companions are her husband “Mr. Stewart,” their three oldest children, and kind-hearted, opinionated neighbor Mrs. O’Shaughnessy. Mr. Haynes (organizer of the hunt) and his friend, Mr. Struble (the cheerful big man of the party) lead the group, and are also joined by physician Dr. Teschall, “a moving-picture man” Mr. Harkrudder, Professor Glenholdt seeking “the tip-end bone of the tail of a brontosaurus” and his students (“two geological fellows” who “talk of nothing but strata and formation”). Also joining the group is Mr. Murry with his tiresome accordion.<br><br>Although some hunting is accomplished on the trip, the overarching focus of Elinore’s letters is on descriptions of awe-inspiring Wyoming scenery and the interesting people she encounters. With her familiar wit and wisdom, Elinore also writes of tragedies and romances she observes during her trip -- that is, whenever Elinore’s effort to observe is not thwarted by “the good mon” Mr. Stewart. In one letter Elinore complains to Mrs. Coney that “Mr. Stewart is the queerest man: instead of letting me enjoy the tableau [the reunion of two long-lost lovers], he solemnly drove on, saying he would not want any one gawking at him if he were the happy man. Anyway, he couldn’t urge Chub [the horse] fast enough to prevent my seeing and hearing what I’ve told you.”<br><br>By the time the adventurers are homeward bound with their supply of elk meat, Elinore is homesick for her youngest child, Junior, at home with his grandmother; Mrs. O’Shaughnessy has taken in two young orphans; and quiet, young Mr. Haynes complains good-naturedly about having to travel along with a rolling nursery.<br><br>Elinore’s letters capture an interesting transition point in history. People traveled by horse and wagon, there were cowboys and cattle stampedes, and medical care was rustic. At the same time, automobiles and modern medicine, archeology and motion picture making were entering the scene and war was commencing in Europe. (Note to more sensitive readers: Elk hunting is described in Chapters 7 and 8.) (Summary by Lynne Carroll)

6Perfect Behavior (Version 2)

By Donald Ogden Stewart

A humorous guide for ladies and gentlemen in all social crises. (Summary by MaryAnn)

7Parody Outline of History

By Donald Ogden Stewart

" Mr. H. G. Wells, in his "Outline of History," was of necessity forced to omit the narration of many of the chief events in the history of these United States. Such omissions I have in this brief volume endeavored to supply. And as American history can possibly best be written by Americans and as we have among us no H. G. Wells, I have imagined an American history as written conjointly by a group of our most characteristic literary figures." - Summary by Donald Ogden Stewart

Welcome back to another exciting collection courtesy of the little known but influential pulp magazine The Thrill Book! In this issue we continue the mystical Egyptian mystery "The Jeweled Ibis" and the daring jungle adventure "In the Shadows of Race" as well as bringing a new full-length raucous and rowdy western "A Hooting, Tooting Son-of-a-Gun" as well as a quality selection of additional stories, poems and other assorted articles. - Summary by Ben Tucker

9Case With Nine Solutions

By Alfred Walter Stewart

This book was written by Alfred Walter Stewart under the pseudonym J.J. Connington: Sir Clinton Driffield is a respected police professional assigned the task of solving a convoluted crime of murder. The case involves a coy temptress, her husband, a secret admirer and a young man with a romantic interest in the temptress. Sir Clinton and his assistant, Inspector Flamborough must whittle down the circumstances and causes of death from nine potential scenarios. The process leads them through scientific clues, messages from a secret source and good old fashioned deductive reasoning. - Summary by Howard Skyman

10Murder In The Maze

By Alfred Walter Stewart

This is another story written by Walter Stewart under the pseudonym of J.J. Connington. His character of Sir Clinton Driffield, the chief constable, is challenged with solving crimes committed in the maze of the Shandon family estate. The Shandons are a successful if flawed set of brothers but all is not well beneath the surface of the privileged lives of their extended family. Jealousies and ambitions cause trouble as they often do. Sir Clinton and his faithful associate Wendover endeavour to unravel a mystery as complex as the maze itself. - Summary by Howard Skyman

11Dangerfield Talisman

By Alfred Walter Stewart

The Dangerfield family resides on an estate steeped in history and tradition. At the centre of this is a much coveted and admired piece of jewelry called the Dangerfield Talisman. A group of guests gathers at the estate for a mixture of business and pleasure. The pleasure provides entertainment for the idle rich while the business is that of a foreigner attempting to purchase the famed article. During their visit the talisman disappears for not the first time in its history and the hunt begins to expose the thief. What this endeavour also exposes are sub-plots involving the different characters in the story and the various dramas and difficulties they encounter. - Summary by Howard Skyman