Displaying similar documents to “Algebraic properties of a family of Jacobi polynomials”

Specializations of one-parameter families of polynomials

Farshid Hajir, Siman Wong (2006)

Annales de l’institut Fourier

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Let K be a number field, and suppose λ ( x , t ) ∈ K [ x , t ] is irreducible over K ( t ) . Using algebraic geometry and group theory, we describe conditions under which the K -exceptional set of λ , i.e. the set of α ∈ K for which the specialized polynomial λ ( x , α ) is K -reducible, is finite. We give three applications of the methods we develop. First, we show that for any fixed n ≥ 10 , all but finitely many K -specializations of the degree n generalized Laguerre polynomial L n ( t ) ( x ) are K -irreducible and have Galois group S n . Second, we study...

A valuation criterion for normal basis generators of Hopf-Galois extensions in characteristic p

Nigel P. Byott (2011)

Journal de Théorie des Nombres de Bordeaux

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Let S / R be a finite extension of discrete valuation rings of characteristic p > 0 , and suppose that the corresponding extension L / K of fields of fractions is separable and is H -Galois for some K -Hopf algebra H . Let 𝔻 S / R be the different of S / R . We show that if S / R is totally ramified and its degree n is a power of p , then any element ρ of L with v L ( ρ ) ≡ - v L ( 𝔻 S / R ) - 1 ( mod n ) generates L as an H -module. This criterion is best possible. These results generalise to the Hopf-Galois situation recent work of G. G. Elder for Galois...

PAC fields over number fields

Moshe Jarden (2006)

Journal de Théorie des Nombres de Bordeaux

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We prove that if K is a number field and N is a Galois extension of ℚ which is not algebraically closed, then N is not PAC over K .

Two remarks on the inverse Galois problem for intersective polynomials

Jack Sonn (2009)

Journal de Théorie des Nombres de Bordeaux

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A (monic) polynomial f ( x ) ∈ ℤ [ x ] is called if the congruence f ( x ) ≡ 0 mod m has a solution for all positive integers m . Call f ( x ) if it is intersective and has no rational root. It was proved by the author that every finite noncyclic solvable group G can be realized as the Galois group over ℚ of a nontrivially intersective polynomial (noncyclic is a necessary condition). Our first remark is the observation that the corresponding result for nonsolvable G reduces to the ordinary inverse Galois...

A classification of the extensions of degree p 2 over ℚ p whose normal closure is a p -extension

Luca Caputo (2007)

Journal de Théorie des Nombres de Bordeaux

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Let k be a finite extension of ℚ p and ℰ k be the set of the extensions of degree p 2 over k whose normal closure is a p -extension. For a fixed discriminant, we show how many extensions there are in ℰ ℚ p with such discriminant, and we give the discriminant and the Galois group (together with its filtration of the ramification groups) of their normal closure. We show how this method can be generalized to get a classification of the extensions in ℰ k .