Displaying similar documents to “Differential Galois Theory for an Exponential Extension of ℂ ( ( z ) ) ”

Random Galois extensions of Hilbertian fields

Lior Bary-Soroker, Arno Fehm (2013)

Journal de Théorie des Nombres de Bordeaux

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Let L be a Galois extension of a countable Hilbertian field K . Although L need not be Hilbertian, we prove that an abundance of large Galois subextensions of L / K are.

Polynomials over Q solving an embedding problem

Nuria Vila (1985)

Annales de l'institut Fourier

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The fields defined by the polynomials constructed in E. Nart and the author in J. Number Theory 16, (1983), 6–13, Th. 2.1, with absolute Galois group the alternating group A n , can be embedded in any central extension of A n if and only if n ≡ 0 ( m o d 8 ) , or n ≡ 2 ( m o d 8 ) and n is a sum of two squares. Consequently, for theses values of n , every central extension of A n occurs as a Galois group over Q .

Quaternion extensions with restricted ramification

Peter Schmid (2014)

Acta Arithmetica

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In any normal number field having Q₈, the quaternion group of order 8, as Galois group over the rationals, at least two finite primes must ramify. The classical example by Dedekind of such a field is extraordinary in that it is totally real and only the primes 2 and 3 are ramified. In this note we describe in detail all Q₈-fields over the rationals where only two (finite) primes are ramified. We also show that, for any integer n>3 and any prime p ≡ 1 ( m o d 2 n - 1 ) , there exist unique real and complex...

Some remarks on Hilbert-Speiser and Leopoldt fields of given type

James E. Carter (2007)

Colloquium Mathematicae

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Let p be a rational prime, G a group of order p, and K a number field containing a primitive pth root of unity. We show that every tamely ramified Galois extension of K with Galois group isomorphic to G has a normal integral basis if and only if for every Galois extension L/K with Galois group isomorphic to G, the ring of integers O L in L is free as a module over the associated order L / K . We also give examples, some of which show that this result can still hold without the assumption that...

Overview of the differential Galois integrability conditions for non-homogeneous potentials

Andrzej J. Maciejewski, Maria Przybylska (2011)

Banach Center Publications

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We report our recent results concerning integrability of Hamiltonian systems governed by Hamilton’s function of the form H = 1 / 2 ∑ i = 1 n p ² i + V ( q ) , where the potential V is a finite sum of homogeneous components. In this paper we show how to find, in the differential Galois framework, computable necessary conditions for the integrability of such systems. Our main result concerns potentials of the form V = V k + V K , where V k and V K are homogeneous functions of integer degrees k and K > k, respectively. We present examples...

Counting discriminants of number fields

Henri Cohen, Francisco Diaz y Diaz, Michel Olivier (2006)

Journal de Théorie des Nombres de Bordeaux

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For each transitive permutation group G on n letters with n ≤ 4 , we give without proof results, conjectures, and numerical computations on discriminants of number fields L of degree n over ℚ such that the Galois group of the Galois closure of L is isomorphic to G .

An explicit integral polynomial whose splitting field has Galois group W ( E 8 )

Florent Jouve, Emmanuel Kowalski, David Zywina (2008)

Journal de Théorie des Nombres de Bordeaux

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Using the principle that characteristic polynomials of matrices obtained from elements of a reductive group G over Q typically have splitting field with Galois group isomorphic to the Weyl group of G , we construct an explicit monic integral polynomial of degree 240 whose splitting field has Galois group the Weyl group of the exceptional group of type E 8 .

An equicharacteristic analogue of Hesselholt's conjecture on cohomology of Witt vectors

Amit Hogadi, Supriya Pisolkar (2013)

Acta Arithmetica

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Let L/K be a finite Galois extension of complete discrete valued fields of characteristic p. Assume that the induced residue field extension k L / k K is separable. For an integer n ≥ 0, let W n ( L ) denote the ring of Witt vectors of length n with coefficients in L . We show that the proabelian group H 1 ( G , W n ( L ) ) n ∈ ℕ is zero. This is an equicharacteristic analogue of Hesselholt’s conjecture, which was proved before when the discrete valued fields are of mixed characteristic.

Noncommutative numerical motives, Tannakian structures, and motivic Galois groups

Matilde Marcolli, Gonçalo Tabuada (2016)

Journal of the European Mathematical Society

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In this article we further the study of noncommutative numerical motives, initiated in [30, 31]. By exploring the change-of-coefficients mechanism, we start by improving some of the main results of [30]. Then, making use of the notion of Schur-finiteness, we prove that the category NNum ( k ) F of noncommutative numerical motives is (neutral) super-Tannakian. As in the commutative world, NNum ( k ) F is not Tannakian. In order to solve this problem we promote periodic cyclic homology to a well-defined...

Quasi-semi-stable representations

Xavier Caruso, Tong Liu (2009)

Bulletin de la Société Mathématique de France

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Fix K a p -adic field and denote by G K its absolute Galois group. Let K ∞ be the extension of K obtained by adding p n -th roots of a fixed uniformizer, and G ∞ ⊂ G K its absolute Galois group. In this article, we define a class of p -adic torsion representations of G ∞ , called. We prove that these representations are “explicitly” described by a certain category of linear algebraic objects. The results of this note should be considered as a first step in the understanding of the structure of quotient...