On the monogenity and Galois group of certain classes of polynomials

Article Type

Research Article

Publication Title

Mathematica Slovaca

Abstract

We say a monic polynomial g(x) ϵ ℤ[x] of degree n is monogenic if g(x) is irreducible over ℚ and {1, θ, ..., θn-1} is a basis for the ring ℤK of integers of number field K = ℚ(θ), where θ is a root of g(x). Let f(x)=xn+cσi=1n(ax)n-iϵZ[x]andF(x)=xn+cσi=1nai-1xn-iϵZ[x] be irreducible polynomials having degree n ≥ 3. In this paper, we provide necessary and sufficient conditions involving only a, c, n for the polynomials f(x) and F(x) to be monogenic. As an application, we also provide a class of polynomials having a non square-free discriminant and Galois group Sn, the symmetric group on n letters.

First Page

1147

Last Page

1154

DOI

10.1515/ms-2024-0082

Publication Date

10-1-2024

Comments

Open Access; Green Open Access

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