Displaying similar documents to “Computing Galois groups by means of Newton polygons”

On the Galois group of generalized Laguerre polynomials

Farshid Hajir (2005)

Journal de Théorie des Nombres de Bordeaux

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Using the theory of Newton Polygons, we formulate a simple criterion for the Galois group of a polynomial to be “large.” For a fixed α ∈ ℚ - ℤ < 0 , Filaseta and Lam have shown that the n th degree Generalized Laguerre Polynomial L n ( α ) ( x ) = ∑ j = 0 n n + α n - j ( - x ) j / j ! is irreducible for all large enough n . We use our criterion to show that, under these conditions, the Galois group of L n ( α ) ( x ) is either the alternating or symmetric group on n letters, generalizing results of Schur for α = 0 , 1 , ± 1 2 , - 1 - n .

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.

Galois realizability of groups of order 64

Helen Grundman, Tara Smith (2010)

Open Mathematics

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This article examines the realizability of groups of order 64 as Galois groups over arbitrary fields. Specifically, we provide necessary and sufficient conditions for the realizability of 134 of the 200 noncyclic groups of order 64 that are not direct products of smaller groups.

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.

Groups of Order 32 as Galois Groups

Michailov, Ivo (2007)

Serdica Mathematical Journal

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2000 Mathematics Subject Classification: 12F12. We find the obstructions to realizability of groups of order 32 as Galois groups over arbitrary field of characteristic not 2. We discuss explicit extensions and automatic realizations as well. This work is partially supported by project of Shumen University

Invariants and differential Galois groups in degree four

Julia Hartmann (2002)

Banach Center Publications

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This note extends the algorithm of [hess] for computing unimodular Galois groups of irreducible differential equations of order four. The main tool is invariant theory.