On classifying Laguerre polynomials which have Galois group the alternating group par
نویسندگان
چکیده
We show that the discriminant of the generalized Laguerre polynomial L n (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 (x) over Q is the alternating group An. 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 (α) n (x) over Q is An is α = n.
منابع مشابه
On the Galois Group of generalized Laguerre polynomials
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 nth degree Generalized Laguerre Polynomial L (α) n (x) = ∑n j=0 ( n+α n−j ) (−x)/j! is irreducible for all large enough n. We use our criterion to show that, under these conditions, the Galois group of L (α) n (x) is...
متن کاملLAGUERRE POLYNOMIALS WITH GALOIS GROUP Am FOR EACH
In 1892, D. Hilbert began what is now called Inverse Galois Theory by showing that for each positive integer m, there exists a polynomial of degree m with rational coefficients and associated Galois group Sm, the symmetric group on m letters, and there exists a polynomial of degree m with rational coefficients and associated Galois group Am, the alternating group on m letters. In the late 1920’...
متن کاملOn the Genus of Generalized Laguerre Polynomials
belong to one of the three family of orthogonal polynomials, the other two being Jacobi and Legendre. In addition to their important roles in mathematical analysis, these polynomials also feature prominently in algebra and number theory. Schur ([7], [8]) pioneered the study of Galois properties of specializations of these orthogonal polynomials, and Feit [1] used them to solve the inverse Galoi...
متن کاملSpecializations of One-parameter Families of Polynomials
Let K be a number field, and let λ(x, t) ∈ K[x, t] be irreducible over K(t). Using algebraic geometry and group theory, we study the set of α ∈ K for which the specialized polynomial λ(x, α) is K-reducible. We apply this to show that for any fixed n ≥ 10 and for any number field K, all but finitely many K-specializations of the degree n generalized Laguerre polynomial L (t) n (x) are K-irreduci...
متن کاملAlgebraic Properties of a Family of Generalized Laguerre Polynomials
We study the algebraic properties of Generalized Laguerre Polynomials for negative integral values of the parameter. For integers r, n ≥ 0, we conjecture that L n (x) = Pn j=0 `n− j+r n− j ́ x / j! is a Q-irreducible polynomial whose Galois group contains the alternating group on n letters. That this is so for r = n was conjectured in the 1950’s by Grosswald and proven recently by Filaseta and T...
متن کامل