Monogenity of the composition of certain polynomials
Streszczenie
We call a monic irreducible polynomial $f(x)\in \mathbb {Z}[x]$ to be monogenic if $\mathbb {Z}[\theta ]$ is the ring of integers of the number field $\mathbb {Q}(\theta )$ where $\theta $ is a root of $f(x)$. Finding the ring of integers of a number field is an important problem in algebraic number theory. In this article, we establish a necessary condition for the monogenity of a composition of two polynomials. In particular, we characterise certain primes dividing the index of the composition $f(x^m+a)$ for $m\geq 2$, $a\in \mathbb {Z}$, provided that $f(x)\in \mathbb {Z}[x]$ is a monic polynomial such that the composition $f(x^m+a)$ is irreducible. As a consequence, we are able to achieve a sufficient condition for the monogenity of $f(x^m+a)$ which in turn enables us to produce a new infinite family of monogenic polynomials.
