Teresa Crespo

On the Galois correspondence theorem in separable Hopf Galois theory

There are fourteen ne gradings on the exceptional Lie algebra e6 over an algebraically closed eld of zero characteristic. We provide their descriptions and a proof that any ne grading is equivalent to one of them. 2010 Mathematics Subject Classication: 17B25, 17B70.There are fourteenfine gradings on the exceptional Lie algebra e6 over an algebraically closed field of zero characteristic. We provide their descriptions and a proof that any fine grading is equivalent to one of them.In this paper we prove mixed norm estimates for Riesz transforms on the group SU(2). From these results vector valued inequalities for sequences of Riesz transforms associated to Jacobi differential operators of different types are deduced.In this paper we introduce new techniques in order to deepen into the structure of a Leavitt path algebra with the aim of giving a description of the center. Extreme cycles appear for the first time; they concentrate the purely infinite part of a Leavitt path algebra and, jointly with the line points and vertices in cycles without exits, are the key ingredients in order to determine the center of a Leavitt path algebra. Our work will rely on our previous approach to the center of a prime Leavitt path algebra.In this paper we present a reformulation of the Galois correspondence theorem of Hopf Galois theory in terms of groups carrying farther the description of Greither and Pareigis. We prove that the class of Hopf Galois extensions for which the Galois correspondence is bijective is larger than the class of almost classically Galois extensions but not equal to the whole class. We show as well that the image of the Galois correspondence does not determine the Hopf Galois structure.We show that the product BMO space can be characterized by iterated commutators of a large class of Calderon-Zygmund operators. This result followsfrom a new proof of boundedness of iterated commutators in terms of the BMO norm of their symbol functions, using Hytonens representation theorem of Calderon-Zygmund operators as averages of dyadic shifts. The proof introduces some new paraproducts which have BMO estimates.In this paper we survey some results on the Dirichlet problem for nonlocal operators of the form. We start from the very basics, proving existence of solutions, maximum principles, and constructing some useful barriers. Then, we focus on the regularity properties of solutions, both in the interior and on the boundary of the domain. In order to include some natural operators L in the regularity theory, we do not assume any regularity on the kernels. This leads to some interesting features that are purely nonlocal, in the sense that they have no analogue for local equations. We hope that this survey will be useful for both novel and more experienced researchers in the field.

The Hopf Galois property in subfield lattices

Let K/k be a finite separable extension, n its degree and its Galois closure. For n ≤ 5, Greither and Pareigis show that all Hopf Galois extensions are either Galois or almost classically Galois and they determine the Hopf Galois character of K/k according to the Galois group (or the degree) of . In this paper we study the case n = 6, and intermediate extensions F/k such that , for degrees n = 4, 5, 6. We present an example of a non almost classically Galois Hopf Galois extension of ℚ of the smallest possible degree and new examples of Hopf Galois extensions. In the last section we prove a transitivity property of the Hopf Galois condition.

Real Liouville Extensions

We characterize linear differential equations defined over a real differential field with a real closed field of constants C, which are solvable by real Liouville functions, as those having a differential Galois group whose identity component is solvable and C-split.

The Valentiner group as Galois group

We obtain the complete set of solutions to the Galois embedding problem given by the Valentiner group as a triple cover of the alternating group A 6 .

Embedding problems with ramification conditions

An interesting point concerning embedding problems over number fields is the knowledge of the ramification set of its solutions. In the present work, we examine the following problem: Let K be a number field, G K its absolute Galois group, ~b an epimorphism from G K onto a finite group G and L[ K the Galois extension associated to qS. We consider the embedding problem: G~ 1 ~A >E ~G ~1 where E is a central extension of G, i.e. A is a trivial G-module, and assume ~b*e = 0 in H2(GK, A), for e the element in H2(G, A) corresponding to E. For the solvable embedding problem (L J K, e), we want to get a solution field M such that the extension M [ K has a reduced ramification set. To study this problem, we take a finite set S of prime ideals of the ring of integers (9 K of the field K containing the prime ideals ramifying in L[ K. We state the embedding problem (L I K, 0 with the additional condition that the solutions are unramified outside S. We denote this new problem by (L I K, e, S). If G s is the Galois group of the maximal extension of K, unramified outside S, we have a commutative diagram:

An effective study of polynomial maps

In this paper, using an effective algorithm, we obtain an equivalent statement to the Jacobian Conjecture. For a polynomial map F on an affine space of dimension n over a field of characteristic 0, we define recursively a finite sequence of polynomial maps. We give an equivalent condition to the invertibility of F as well as a formula for F−1 in terms of this finite sequence of polynomial maps. Some examples illustrate the effective aspects of our approach.

Galois Correspondence Theorem for Picard-Vessiot Extensions

For a homogeneous linear differential equation defined over a differential field K, a Picard-Vessiot extension is a differential field extension of K differentially generated by a fundamental system of solutions of the equation and not adding constants. When K has characteristic 0 and the field of constants of K is algebraically closed, it is well known that a Picard-Vessiot extension exists and is unique up to K-differential isomorphism. In this case the differential Galois group is defined as the group of K-differential automorphisms of the Picard-Vessiot extension and a Galois correspondence theorem is settled. Recently, Crespo, Hajto and van der Put have proved the existence and unicity of the Picard-Vessiot extension for formally real (resp. formally p-adic) differential fields with a real closed (resp. p-adically closed) field of constants. This result widens the scope of application of Picard-Vessiot theory beyond the complex field. It is then necessary to give an accessible presentation of Picard-Vessiot theory for arbitrary differential fields of characteristic zero which eases its use in physical or arithmetic problems. In this paper, we give such a presentation avoiding both the notions of differential universal extension and specializations used by Kolchin and the theories of schemes and Hopf algebras used by other authors. More precisely, we give an adequate definition of the differential Galois group as a linear algebraic group and a new proof of the Galois correspondence theorem for a Picard-Vessiot extension of a differential field with non algebraically closed field of constants, which is more elementary than the existing ones.

On vectorial polynomials and coverings in characteristic 3

For K a field containing the finite field F g we give explicitly the whole family of Galois extensions of K with Galois group 2S 4 * Q 8 or 2S 4 * D 8 and determine the discriminant of such an extension.

2₄*₈-extensions in characteristic 3

We present an explicit construction of the complete family of Galois extensions of a field K of characteristic 3 with Galois group the central product 2S 4 * Qs of a double cover 2S 4 of the symmetric group S 4 and the quaternion group Q 8 , containing a given S 4 -extension of the field K.

2S4 ∗ Q8-extensions

Let 2S 4 * Q 8 be the central product of a double cover 2S 4 of the symmetric group S 4 and the quaternion group Q 8. We consider the Galois embedding problem given by 2S 4 * Q 8 as a double cover of the direct product S 4 × V 4 of the symmetric group S 4 and the Klein group V 4 over a field K of characteristic different from 2. If 2 or −2 is a square in K, we give a general formula for the solutions to this embedding problem, whenever it is solvable, in terms of quadratic forms. This result answers a question raised by Abhyankar.