Displaying similar documents to “Arithmetic Properties of Generalized Rikuna Polynomials”

On classifying Laguerre polynomials which have Galois group the alternating group

Pradipto Banerjee, Michael Filaseta, Carrie E. Finch, J. Russell Leidy (2013)

Journal de Théorie des Nombres de Bordeaux

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We show that the discriminant of the generalized Laguerre polynomial L n ( α ) ( x ) is a non-zero square for some integer pair ( n , α ) , with n 1 , if and only if ( n , α ) belongs to one of 30 explicitly given infinite sets of pairs or to an additional finite set of pairs. As a consequence, we obtain new information on when the Galois group of L n ( α ) ( x ) over is the alternating group A n . For example, we establish that for all but finitely many positive integers n 2 ( mod 4 ) , the only α for which the Galois group of L n ( α ) ( x ) over is A n is...

When is the order generated by a cubic, quartic or quintic algebraic unit Galois invariant: three conjectures

Stéphane R. Louboutin (2020)

Czechoslovak Mathematical Journal

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Let ε be an algebraic unit of the degree n 3 . Assume that the extension ( ε ) / is Galois. We would like to determine when the order [ ε ] of ( ε ) is Gal ( ( ε ) / ) -invariant, i.e. when the n complex conjugates ε 1 , , ε n of ε are in [ ε ] , which amounts to asking that [ ε 1 , , ε n ] = [ ε ] , i.e., that these two orders of ( ε ) have the same discriminant. This problem has been solved only for n = 3 by using an explicit formula for the discriminant of the order [ ε 1 , ε 2 , ε 3 ] . However, there is no known similar formula for n > 3 . In the present paper, we put forward and...

Fields of moduli of three-point G -covers with cyclic p -Sylow, II

Andrew Obus (2013)

Journal de Théorie des Nombres de Bordeaux

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We continue the examination of the stable reduction and fields of moduli of G -Galois covers of the projective line over a complete discrete valuation field of mixed characteristic ( 0 , p ) , where G has a p -Sylow subgroup P of order p n . Suppose further that the normalizer of P acts on P via an involution. Under mild assumptions, if f : Y 1 is a three-point G -Galois cover defined over ¯ , then the n th higher ramification groups above p for the upper numbering of the (Galois closure of...

On the compositum of all degree d extensions of a number field

Itamar Gal, Robert Grizzard (2014)

Journal de Théorie des Nombres de Bordeaux

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We study the compositum k [ d ] of all degree d extensions of a number field k in a fixed algebraic closure. We show k [ d ] contains all subextensions of degree less than d if and only if d 4 . We prove that for d > 2 there is no bound c = c ( d ) on the degree of elements required to generate finite subextensions of k [ d ] / k . Restricting to Galois subextensions, we prove such a bound does not exist under certain conditions on divisors of d , but that one can take c = d when d is prime. This question was inspired by work of...

On standard norm varieties

Nikita A. Karpenko, Alexander S. Merkurjev (2013)

Annales scientifiques de l'École Normale Supérieure

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Let  p be a prime integer and F a field of characteristic 0 . Let  X be theof a symbol in the Galois cohomology group H n + 1 ( F , μ p n ) (for some n 1 ), constructed in the proof of the Bloch-Kato conjecture. The main result of the paper affirms that the function field F ( X ) has the following property: for any equidimensional variety Y , the change of field homomorphism CH ( Y ) CH ( Y F ( X ) ) of Chow groups with coefficients in integers localized at  p is surjective in codimensions < ( dim X ) / ( p - 1 ) . One of the main ingredients of the proof is a computation...

Realizable Galois module classes over the group ring for non abelian extensions

Nigel P. Byott, Bouchaïb Sodaïgui (2013)

Annales de l’institut Fourier

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Given an algebraic number field k and a finite group Γ , we write ( O k [ Γ ] ) for the subset of the locally free classgroup Cl ( O k [ Γ ] ) consisting of the classes of rings of integers O N in tame Galois extensions N / k with Gal ( N / k ) Γ . We determine ( O k [ Γ ] ) , and show it is a subgroup of Cl ( O k [ Γ ] ) by means of a description using a Stickelberger ideal and properties of some cyclic codes, when k contains a root of unity of prime order p and Γ = V C , where V is an elementary abelian group of order p r and C is a cyclic group of order m > 1 acting faithfully...

Coincidence for substitutions of Pisot type

Marcy Barge, Beverly Diamond (2002)

Bulletin de la Société Mathématique de France

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Let ϕ be a substitution of Pisot type on the alphabet 𝒜 = { 1 , 2 , ... , d } ; ϕ satisfies theif for every i , j 𝒜 , there are integers k , n such that ϕ n ( i ) and ϕ n ( j ) have the same k -th letter, and the prefixes of length k - 1 of ϕ n ( i ) and ϕ n ( j ) have the same image under the abelianization map. We prove that the strong coincidence condition is satisfied if d = 2 and provide a partial result for d 2 .