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A Class of Trinomials with Galois Group S_n

Anuj Bishnoi

Department of Mathematics, Panjab University, Chandigarh-160014, India

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and

Sudesh K. Khanduja

Department of Mathematics, Panjab University, Chandigarh-160014, India

Corresponding author.

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https://doi.org/10.1142/S1005386712000764Cited by:5 (Source: Crossref)

Abstract

A well known result of Schur states that if n is a positive integer and a₀, a₁,…,a_n are arbitrary integers with a₀a_n coprime to n!, then the polynomial is irreducible over the field ℚ of rational numbers. In case each a_i = 1, it is known that the Galois group of f_n(x) over ℚ contains A_n, the alternating group on n letters. In this paper, we extend this result to a larger class of polynomials f_n(x) which leads to the construction of trinomials of degree n for each n with Galois group S_n, the symmetric group on n letters.

The financial support by CSIR (grant 09/135(0539)/2008-EMR-I) and National Board for Higher Mathematics, Mumbai is gratefully acknowledged.

Keywords:

AMSC: 12E05, 11R32, 12J25

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