Abstract
Let f(x) ∈ ℤ[x] be monic and irreducible over ℚ, with deg(f) = n. Let K = ℚ(θ), where f(θ) = 0, and let ℤK denote the ring of integers of K. We say f(x) is monogenic if {1, θ, θ2, …, θn−1} is a basis for ℤK. Otherwise, f(x) is called non-monogenic. In this article, we give necessary and sufficient conditions for a certain class of polynomials to be monogenic. Using these conditions allows us to generate infinite families of non-monogenic polynomials. In particular, for quadrinomials our results show that there exist infinitely many primes p ≥ 3, and integers t ≥ 1 coprime to p, such that f(x) = xp − 2ptxp−1 + p2t2xp−2 + 1 is non-monogenic. Finally, we illustrate this situation with an explicit example.
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(Communicated by István Gaál)
Acknowledgement
The author thanks the anonymous referees for the helpful suggestions.
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