We explore the constructive characterization of all maximal ideals in ℤ[X], which serves as a case study for the concept of the "Material Interpretation". Starting with a classical proof, we demonstrate how it can be transformed into a constructive proof, potentially leading to stronger statements. The Material Interpretation provides an alternative proof interpretation, transforming statements of the form A → B into ¬A ∨ B, where ¬A represents a constructively stronger form of the negation of A. We will discuss the theoretical foundations of this approach as well as its formalization in the proof assistant Agda, which is still a work in progress. Furthermore we highlight Agda's capabilities for automating the Material Interpretation and share improvements to its library through the addition of valuable theorems.