Python programming and algebra: Some special elements in Gaussian integers modulo a prime

We consider the complexification of a commutative ring with unity and specialize this construction to Z_p[i], with p a prime of the form 4k+1. Since this ring is commutative with unity and is not a field, it is feasible to study various classes of special elements such as invertibles, zero-divisors, idempotents, and nilpotents. The method for this study consists of developing computer programs in Python, through which the lists of special elements in Z_p[i] are generated for different values of p. The patterns that characterize these lists are sought, in addition to the cardinality of each of these sets. Subsequently, conjectures of mathematical type are stated for each of these classes of elements, which reflect the observed patterns and properties. Finally, formal mathematical proofs of all the conjectures found are made based on various concepts and results of the theory of numbers, groups, and rings. Thus, we show that Python programming, properly used as part of a method, becomes an important tool to identify patterns, properties, and characteristics of several abstract concepts, typical of algebra.