$2$-superirreducibility of univariate polynomials over $\mathbb{Q}$ and $\mathbb{Z}$

TL;DR

This paper studies the stability of polynomial irreducibility over rationals and integers under quadratic substitutions, providing classifications for degrees 2, 3, and 4, and exploring higher degrees with conjectures.

Contribution

It establishes the non-existence of 2-superirreducible polynomials of degree 2 and 3, and constructs examples and conjectures for degree 4 and higher.

Findings

Degree 2 and 3 polynomials are not 2-superirreducible.

Constructed families of degree 4 polynomials with different irreducibility behaviors.

Proposed conjectures for higher degree cases.

Abstract

This paper investigates whether or not polynomials that are irreducible over $\mathbb{Q}$ and $\mathbb{Z}$ can remain irreducible under substitution by all quadratic polynomials. It answers this question in the negative in the degree 2 and 3 cases and provides families of examples in both the affirmative and the negative categories in the degree 4 case. Finally this paper explores what happens in higher degree cases, providing a family of examples in the negative category and offering a conjectured family for the positive category.