In this article we construct some Galois extensions L∕K with finite Galois groups and such that |Gal(L∕K)|>L:K. Using an analog of the Noether method, we explain how to obtain, with a fixed center, ...such a Galois curiosity with a Galois group as large as we want.
Résumé : Dans cet article, nous construisons une extension galoisienne L∕K à groupe de Galois fini et telle que |Gal(L∕K)|>L:K. En utilisant un analogue non commutatif de la méthode de Noether, nous expliquons ensuite comment, à centre fixé, l'on peut construire une telle curiosité galoisienne avec un groupe aussi gros que l'on veut.
Mots clés : Corps gauches; théorie de Galois.
We apply the differential Galois theory for difference equations developed by Hardouin and Singer to compute the differential Galois group for a second-order linear q-difference equation with ...rational function coefficients. This Galois group encodes the possible polynomial differential relations among the solutions of the equation. We apply our results to compute the differential Galois groups of several concrete q-difference equations, including for the colored Jones polynomial of a certain knot.
In 2, the author claims that the fields Q(D4∞) defined in the paper and the compositum of all D4 extensions of Q coincide. The proof of this claim depends on a misreading of a celebrated result by ...Shafarevich. The purpose is to salvage the main results of 2. That is, the classification of torsion structures of E defined over Q when base changed to the compositum of all D4 extensions of Q main results of 2. All the main results in 2 are still correct except that we are no longer able to prove that these two fields are equal.
Let E/Q be an elliptic curve and let Q(D4∞) be the compositum of all extensions of Q whose Galois closure has Galois group isomorphic to a quotient of a subdirect product of a finite number of ...transitive subgroups of D4. In this article we first show that Q(D4∞) is in fact the compositum of all D4 extensions of Q and then we prove that the torsion subgroup of E(Q(D4∞)) is finite and determine the 24 possibilities for its structure. We also give a complete classification of the elliptic curves that have each possible torsion structure in terms of their j-invariants.
We compute the Galois group of a polynomial whose roots are determined by the critical points of a scalar potential in type IIB compactifications. We focus our study on certain perturbative models ...where it is feasible to construct a de Sitter vacuum within the effective theory by introducing non-geometric fluxes, D-branes or non-BPS states. Our findings clearly show that all de Sitter vacua derived from lifting AdS stable vacua are associated with an unsolvable Galois group. This suggests a deeper connection between the fundamental principles of Galois theory and its applications in the construction of dS vacua.
•In the present manuscript it is demonstrated that all de Sitter vacua from uplifting Anti-de Sitter stable vacua have associated a unsolvable Galois groups.•Due to their association with unsolvable Galois groups, all dS vacua lack analytic solutions.•The findings significantly impact the string theory landscape, offering a new perspective on the complexity of scalar potentials and its critical points as well as the stability of dS spaces.•Utilizing Galois theory to explore scalar potentials introduces a novel methodology, bridging advanced mathematical theories with string phenomenology and opening new research avenues for dS spaces in string theory.
In this paper we revisit the following inverse problem: given a curve invariant under an irreducible finite linear algebraic group, can we construct an ordinary linear differential equation whose ...Schwarz map parametrizes it? We present an algorithmic solution to this problem under the assumption that we are given the function field of the quotient curve. The result provides a generalization and an efficient implementation of the solution to the inverse problem exposed by van der Put et al. (2020). As an application, we show that there is no hypergeometric equation with projective differential Galois group isomorphic to H72, thus completing Beukers and Heckman’s answer (Beukers and Heckman, 1989) to the question of which irreducible finite subgroup of PSL3(ℂ) are the projective monodromy of a hypergeometric equation.
We relate two different proposals to extend the étale topology into homotopy theory, namely via the notion of finite cover introduced by Mathew and via the notion of separable commutative algebra ...introduced by Balmer. We show that finite covers are precisely those separable commutative algebras with underlying dualizable module, which have a locally constant and finite degree function. We then use Galois theory to classify separable commutative algebras in numerous categories of interest. Examples include the category of modules over a connective E∞-ring R which is either connective or even periodic with π0(R) regular Noetherian in which 2 acts invertibly, the stable module category of a finite group of p-rank one and the derived category of a qcqs scheme.
We investigate finite sets of rational functions {f1,f2,…,fr} defined over some number field K satisfying that any t0∈K is a Kp-value of one of the functions fi for almost all primes p of K. We give ...strong necessary conditions on the shape of functions appearing in a minimal set with this property, as well as numerous concrete examples showing that these necessary conditions are in a way also close to sufficient. We connect the problem to well-studied concepts such as intersective polynomials and arithmetically exceptional functions.
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