Displaying similar documents to “Two remarks on the inverse Galois problem for intersective polynomials”

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 .

Differential Galois Theory for an Exponential Extension of ( ( z ) )

Magali Bouffet (2003)

Bulletin de la Société Mathématique de France

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In this paper we study the formal differential Galois group of linear differential equations with coefficients in an extension of ( ( z ) ) by an exponential of integral. We use results of factorization of differential operators with coefficients in such a field to give explicit generators of the Galois group. We show that we have very similar results to the case of ( ( z ) ) .

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...

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...

Why quintic polynomial equations are not solvable in radicals

Křížek, Michal, Somer, Lawrence

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We illustrate the main idea of Galois theory, by which roots of a polynomial equation of at least fifth degree with rational coefficients cannot general be expressed by radicals, i.e., by the operations + , - , · , : , and · n . Therefore, higher order polynomial equations are usually solved by approximate methods. They can also be solved algebraically by means of ultraradicals.

Reduction and specialization of polynomials

Pierre Dèbes (2016)

Acta Arithmetica

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We show explicit forms of the Bertini-Noether reduction theorem and of the Hilbert irreducibility theorem. Our approach recasts in a polynomial context the geometric Grothendieck good reduction criterion and the congruence approach to HIT for covers of the line. A notion of “bad primes” of a polynomial P ∈ ℚ[T,Y] irreducible over ℚ̅ is introduced, which plays a central and unifying role. For such a polynomial P, we deduce a new bound for the least integer t₀ ≥ 0 such that P(t₀,Y) is...

Galois co-descent for étale wild kernels and capitulation

Manfred Kolster, Abbas Movahhedi (2000)

Annales de l'institut Fourier

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Let F be a number field with ring of integers o F . For a fixed prime number p and i 2 the étale wild kernels W K 2 i - 2 e ´ t ( F ) are defined as kernels of certain localization maps on the i -fold twist of the p -adic étale cohomology groups of spec o F [ 1 p ] . These groups are finite and coincide for i = 2 with the p -part of the classical wild kernel W K 2 ( F ) . They play a role similar to the p -part of the p -class group of F . For class groups, Galois co-descent in a cyclic extension L / F is described by the ambiguous class formula given...

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.

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 .

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...