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Showing 1 to 20 of 30 for “"Galois group"”.
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The Absolute Galois Group of the Rationals, Grothendieck's Dessin D'Enfants, and Galois Invariants
… a highly non-trivial action of the absolute Galois group of the rationals on a collection of relatively simple combinatorial objects. We then analyze recent work by Girondo, Gonzalez-Diez, Hidalgo, and Jones which provides two new Galois invariants for dessins called ''Zapponi orientability" …
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On the Galois Group of the 2-Class Field Towers of Some Imaginary Quadratic Fields
… field. These patterns give rise to a class of group extensions that includes each of the 3 groups. We conjecture subgroup and quotient group properties of these extensions.
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On the Complexity of Computing Galois Groups of Differential Equations
<p>The differential Galois group is an analogue for a linear differential equation of the classical Galois group for a polynomial equation. An important application of the differential Galois group is that a linear differential equation can be solved by integrals, exponentials and algebraic …
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Topics in Inverse Galois Theory
Galois theory, the study of the structure and symmetry of a polynomial or associated field extension, is a standard tool for showing the insolvability of a quintic equation by radicals. On the other hand, the Inverse Galois Problem, given a finite group G, find a finite extension of the rational …
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Massey products in the Galois cohomology of number fields
We study the relation structure of the Galois group of the maximal outside a finite set of primes unramified p-extension of Q and of the Galois group of the p-class field tower of a quadratic number field. This relation structure is described in terms of Massey products in the Galois cohomology of …
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Lifting Galois Representations in a Conjecture of Figueiredo
… odd irreducible representations of the absolute Galois group of Q and modular forms. Letting M be an imaginary quadratic field, L.M. Figueiredo gave a related conjecture concerning degree 2 irreducible representations of the absolute Galois group of M and their correspondence to homology classes. …
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Toward a deformation theory for Galois representations of function fields
… over a finite field of the absolute Galois group of a function field K. In this case, the characterization of abelian p-power extensions of fields of characteristic p can be extended and refined to allow only restricted ramification at the places of K, and can be a tool for analyzing …
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On the Inverse Problem in Differential Galois Theory
Differential Galois theory generalizes the usual Galois theory for polynomials to differential equations. There is the notion of a splitting field (Picard-Vessiot extension) of a differential equation, and the differential Galois group is the group of automorphisms of this extension which fix the …
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Calculation of Galois groups
<p>In the 19th Century Galois developed a method for determining whether an equation is solvable. It relied on the close relationship between fields and their automorphism group. This paper is a survey of the techniques of Galois theory. After presenting the main results of elementary Galois theory …
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Quintic Abelian Fields
… in terms of their conductor and a certain Galois group. From these, a generating polynomial and its roots and an integral basis are computed. A method for finding the fundamental units, regulators and class numbers is then developed. Tables listing the coefficients of a generating …
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Non-commutative Iwasawa theory of elliptic curves at primes of multiplicative reduction
… false Tate curve extensions F[infinity], with Galois group G = Gal(F[infinity]/Q). I show how the p[infinity]-Selmer group of E over F[infinity] controls the p[infinity]-Selmer rank growth within the false Tate curve extension, and how it is connected to the root numbers of E twisted by …
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On A Conjecture of Pal Turan and Investigations Into Galois Groups of Generalized Laguerre Polynomials
… <p>In the second half we investigate the Galois groups associated with the generalized Laguerre polynomials. We are able to classify Laguerre polynomials with the alternating group as the Galois group. We further compute the Galois groups in certain particular cases.</p>
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A Survey of Galois Theory
In this paper, we will see a summary of Galois theory and some results utilized in Algebra. We state and prove the Fundamental Theorem of Galois theory and then work a few examples to show its application to the determination of lattices of groups and fields. In Chapter 2, we prove the theorem …
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Towards a topological proof of Wright's Theorem
… of degree-n extensions of a number field with Galois group permutation-isomorphic to G. Using the function field analogy, a similar conjecture can be made for finite extensions of F_q(t), where q is a power of a prime. In fact, a similar conjecture has been proven by Wright (1989), in the case …
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On the Modularity of Higher-Dimensional Varieties
… 30, and sigma is the nontrivial element in the Galois group Gal( Q&parl0;5&parr0;/ Q ).
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Arboreal representations, sectional monodromy groups, and abelian varieties over finite fields
… first part studies arboreal representations of Galois groups - an arithmetic dynamics analogue of Tate modules - and proves some large image results, in particular confirming a conjecture of Odoni. Given a field K, a separable polynomial [mathematical expression], and an element [mathematical …
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Higher-Order Lucas Sequences and Dickson Polynomials
… of the extension is interpreted in terms of the Galois group of h. Lastly, we examine the case where F is finite, and offer a formula for M of order 5.
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Local-to-Global extensions for wildly ramified covers of curves
Given a Galois cover of curves X --> Y with Galois group G which is totally ramified at a point x and unramified elsewhere, restriction to the punctured formal neighborhood of x induces a Galois extension of Laurent series rings k((u))/k((t)). If we fix a base curve Y, we can ask when a Galois …
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On the slope filtration of [phi]-modules over the Robba ring
Given a p-adic representation of the Galois group of a local field, we show that its Galois cohomology can be computed using the associated étale ([phi], [Gamma])-module over the Robba ring; this is a variant of a result of Herr. We then establish analogues, for not necessarily étale (([phi], …
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On the Galois module structure of the units and ray classes of a real abelian number field
We study the Galois module structure of the ideal ray class group and the group of units of a real abelian number field. Specifically, we derive explicit annihilators of the ideal ray class groups in the vein of the classical Stickelberger theorems. This is made possible by generalizing a theorem …
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