Year: 2025
Type: article
Abstract: The main purpose of this article is to extract infinite families of polynomials of the form $$ t^n + c(at^k + b)^m \in \mathbb {Z}[t] $$ , where $$ k \ge 2 $$ and the degree $$ n > mk + 1 $$ is fixed, such that the discriminants are not necessarily squarefree, yet the polynomials are either monogenic or possess the full symmetric group as their Galois group. In particular, we establish lower bounds on the number of monogenic polynomials of this form for the cases $$(i)$$ $$ m = 1 $$ , and $$(ii)$$ $$ b = 1 $$ when $$ m \ge 2 $$ . Furthermore, for any prime number $$ p $$ , we provide a lower bound on the number of monogenic polynomials of the form $$ t^p + c(at^k + b)^m \in \mathbb {Z}[t] $$ of degree $$ p $$ , whose Galois group over $$ \mathbb {Q} $$ is isomorphic to the symmetric group $$ S_p $$ .
Source: The Ramanujan Journal
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $2,990
OpenAlex ID: W4410701201