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

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.