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

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.