Isaac Agyei
Ohio University
Title
Factorization of Irreducible Polynomials over Rational and Finite Fields
Abstract
We explore an extension of the algebra of polynomials on a single variable, recently introduced by L\'opez-Permouth and Pallone, called algebras of $m$-nomials and entangled polynomials. We are motivated by the fact that there may exist non-trivial factorizations of irreducible polynomials within the context of entangled polynomials. We are interested in the notion of a positive integer $n$ $(n\in \mathbb{Z}^{+})$ such that $f(x)$ $n-$nomial is reducible. The valence is the smallest $n \times n$ matrix such that it is possible to write the $f(x)$ $n-$nomial as the product of two nonunits in $K^n[x]$.
The valence can also be infinity if no such $n$ exists. It was shown in Ashley and Lopez's paper that linear polynomials have infinite valence. We explore possible ways in which the degree of a polynomial can influence its valence. We present, as a key element, how the determinant of a polynomial, when viewed as an $n$-nomial via a finitistic representation also introduced in Ashley and Lopez's Paper, helps determine the moment when irreducibility is lost. In this study, we focus on cases in which the underlying fields are rational or finite.
Zoom Link: https://osu.zoom.us/j/98492698311?pwd=yimtIdmH6TKoCbW9h7kQssTJSLEU5V.1