Indecomposable polynomials and their spectrum

TL;DR

This paper investigates the properties of indecomposable polynomials, focusing on their spectrum behavior under various algebraic operations and comparing indecomposability over different fields, with implications for finite field polynomials.

Contribution

It provides new insights into how indecomposability and spectrum properties are preserved or change under reduction, specialization, and field extensions.

Findings

Spectrum behavior under reduction and specialization analyzed

Indecomposability properties compared over fields and algebraic closures

Quantitative results on decomposable polynomials over finite fields

Abstract

We address some questions concerning indecomposable polynomials and their spectrum. How does the spectrum behave via reduction or specialization, or via a more general ring morphism? Are the indecomposability properties equivalent over a field and over its algebraic closure? How many polynomials are decomposable over a finite field?