scholarly journals Rational Trinomials with the Alternating Group as Galois Group

2001 ◽  
Vol 90 (1) ◽  
pp. 113-129 ◽  
Author(s):  
Alain Hermez ◽  
Alain Salinier
Keyword(s):  
Galois Group ◽  
Alternating Group

Infinite Families of A4-Sextic Polynomials

2014 ◽  
Vol 57 (3) ◽  
pp. 538-545 ◽  
Author(s):  
Joshua Ide ◽  
Lenny Jones
Keyword(s):  
Galois Group ◽  
Alternating Group

AbstractIn this article we develop a test to determine whether a sextic polynomial that is irreducible overℚhas Galois group isomorphic to the alternating groupA4. This test does not involve the computation of resolvents, and we use this test to construct several infinite families of such polynomials.

A Class of Trinomials with Galois Group Sn

2012 ◽  
Vol 19 (spec01) ◽  
pp. 905-911 ◽  
Author(s):  
Anuj Bishnoi ◽  
Sudesh K. Khanduja
Keyword(s):  
Positive Integer ◽  
Symmetric Group ◽  
Galois Group ◽  
Rational Numbers ◽  
Alternating Group ◽  
Polynomial Formula

A well known result of Schur states that if n is a positive integer and a0, a1,…,an are arbitrary integers with a0an coprime to n!, then the polynomial [Formula: see text] is irreducible over the field ℚ of rational numbers. In case each ai = 1, it is known that the Galois group of fn(x) over ℚ contains An, the alternating group on n letters. In this paper, we extend this result to a larger class of polynomials fn(x) which leads to the construction of trinomials of degree n for each n with Galois group Sn, the symmetric group on n letters.

Tame kernels and second regulators of number fields and their subfields

2013 ◽  
Vol 12 (1) ◽  
pp. 137-165 ◽  
Author(s):  
Jerzy Browkin ◽  
Herbert Gangl
Keyword(s):  
Zeta Function ◽  
Galois Group ◽  
Number Field ◽  
Dihedral Group ◽  
Number Fields ◽  
Numerical Results ◽  
Alternating Group ◽  
Dedekind Zeta Function ◽  
Tame Kernel ◽  
Tame Kernels

AbstractAssuming a version of the Lichtenbaum conjecture, we apply Brauer-Kuroda relations between the Dedekind zeta function of a number field and the zeta function of some of its subfields to prove formulas relating the order of the tame kernel of a number field F with the orders of the tame kernels of some of its subfields. The details are given for fields F which are Galois over ℚ with Galois group the group ℤ/2 × ℤ/2, the dihedral group D2p; p an odd prime, or the alternating group A4. We include numerical results illustrating these formulas.

scholarly journals Algebraic Properties of a Family of Generalized Laguerre Polynomials

2009 ◽  
Vol 61 (3) ◽  
pp. 583-603 ◽  
Author(s):  
Farshid Hajir
Keyword(s):  
Galois Group ◽  
Laguerre Polynomials ◽  
Irreducible Polynomial ◽  
Main Tool ◽  
Alternating Group ◽  
Newton Polygons ◽  
Generalized Laguerre Polynomials ◽  
Algebraic Properties

Abstract.We study the algebraic properties of Generalized Laguerre Polynomials for negative integral values of the parameter. For integers r, n ≥ 0, we conjecture that 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 Trifonov. It follows fromrecent work of Hajir andWong that the conjecture is true when r is large with respect to n ≥ 5. Here we verify it in three situations: (i) when n is large with respect to r, (ii) when r ≤ 8, and (iii) when n ≤ 4. The main tool is the theory of p-adic Newton Polygons.

scholarly journals On classifying Laguerre polynomials which have Galois group the alternating group

2013 ◽  
Vol 25 (1) ◽  
pp. 1-30 ◽  
Author(s):  
Pradipto Banerjee ◽  
Michael Filaseta ◽  
Carrie E. Finch ◽  
J. Russell Leidy
Keyword(s):  
Galois Group ◽  
Laguerre Polynomials ◽  
Alternating Group

WILD RAMIFICATION IN TRINOMIAL EXTENSIONS AND GALOIS GROUPS

2020 ◽  
Vol 63 (1) ◽  
pp. 106-120
Author(s):  
BOUALEM BENSEBAA ◽  
ABBAS MOVAHHEDI ◽  
ALAIN SALINIER
Keyword(s):  
Symmetric Group ◽  
Galois Group ◽  
Galois Groups ◽  
Alternating Group ◽  
Prime Degree ◽  
Wild Ramification ◽  
Wide Range

AbstractIt is proven that, for a wide range of integers s (2 < s < p − 2), the existence of a single wildly ramified odd prime l ≠ p leads to either the alternating group or the full symmetric group as Galois group of any irreducible trinomial Xp + aXs + b of prime degree p.

scholarly journals The Galois group of f(xr)

1984 ◽  
Vol 25 (1) ◽  
pp. 75-91 ◽  
Author(s):  
S. D. Cohen ◽  
W. W. Stothers
Keyword(s):  
Symmetric Group ◽  
Galois Group ◽  
Irreducible Polynomial ◽  
Alternating Group

Let f(x) be an irreducible polynomial of degree n with coefficients in a field L and r be an integer prime to the characteristic of L. The object of this paper is to describe the galois group g of f(xr) over L when the galois group G of f(x) itself over L is either the full symmetric group Snor the alternating group An. We shall call f standard if G = Sn or An with |G|>2.

scholarly journals The differential Galois group of the rational function field

2021 ◽  
Vol 381 ◽  
pp. 107605
Author(s):  
Annette Bachmayr ◽  
David Harbater ◽  
Julia Hartmann ◽  
Michael Wibmer
Keyword(s):  
Galois Group ◽  
Rational Function ◽  
Function Field ◽  
Differential Galois Group ◽  
Rational Function Field

scholarly journals The Galois group of Xp+aXp−1+a

2009 ◽  
Vol 129 (4) ◽  
pp. 824-830 ◽  
Author(s):  
B. Bensebaa ◽  
A. Movahhedi ◽  
A. Salinier
Keyword(s):  
Galois Group
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