Displaying similar documents to “Ambiguity Theory, Old and New”

Charles Hermite’s stroll through the Galois fields

Catherine Goldstein (2011)

Revue d'histoire des mathématiques

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Although everything seems to oppose the two mathematicians, Charles Hermite’s role was crucial in the study and diffusion of Évariste Galois’s results in France during the second half of the nineteenth century. The present article examines that part of Hermite’s work explicitly linked to Galois, the reduction of modular equations in particular. It shows how Hermite’s mathematical convictions—concerning effectiveness or the unity of algebra, analysis and arithmetic—shaped his interpretation...

Quantum-classical interactions and galois type extensions

Władysław Marcinek (2003)

Banach Center Publications

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An algebraic model for the relation between a certain classical particle system and the quantum environment is proposed. The quantum environment is described by the category of possible quantum states. The initial particle system is represented by an associative algebra in the category of states. The key new observation is that particle interactions with the quantum environment can be described in terms of Hopf-Galois theory. This opens up a possibility to use quantum groups in our model...

Remarks on the intrinsic inverse problem

Daniel Bertrand (2002)

Banach Center Publications

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The intrinsic differential Galois group is a twisted form of the standard differential Galois group, defined over the base differential field. We exhibit several constraints for the inverse problem of differential Galois theory to have a solution in this intrinsic setting, and show by explicit computations that they are sufficient in a (very) special situation.

Differential equations and algebraic transcendents: french efforts at the creation of a Galois theory of differential equations 1880–1910

Tom Archibald (2011)

Revue d'histoire des mathématiques

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A “Galois theory” of differential equations was first proposed by Émile Picard in 1883. Picard, then a young mathematician in the course of making his name, sought an analogue to Galois’s theory of polynomial equations for linear differential equations with rational coefficients. His main results were limited by unnecessary hypotheses, as was shown in 1892 by his student Ernest Vessiot, who both improved Picard’s results and altered his approach, leading Picard to assert that his lay...

On the inverse problem of Galois theory.

Núria Vila (1992)

Publicacions Matemàtiques

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The problem of the construction of number fields with Galois group over Q a given finite groups has made considerable progress in the recent years. The aim of this paper is to survey the current state of this problem, giving the most significant methods developed in connection with it.

Realizability and automatic realizability of Galois groups of order 32

Helen Grundman, Tara Smith (2010)

Open Mathematics

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This article provides necessary and sufficient conditions for each group of order 32 to be realizable as a Galois group over an arbitrary field. These conditions, given in terms of the number of square classes of the field and the triviality of specific elements in related Brauer groups, are used to derive a variety of automatic realizability results.