Narrow Results

Let $n\in \mathbb{Z}$ with $n\ge 3$. Let $S_n$ and $A_n$ denote, respectively, the symmetric group and alternating group on $n$ letters. Let $m$ be an indeterminate, and define

For each integer $n\ge 2$, we identify new infinite families of monogenic trinomials $f(x)=x^n+A{x}^m+B$ with non-squarefree discriminant, many of which have small Galois group. Moreover, in certain situations when $A=B\ge 2$ with fixed $n$ and $m$, we produce asymptotics on the number of such trinomials with $A\le X$.