Surprising Mathematics Using a Computer Algebra System

In the course of studying some idempotents of the group algebra of the symmetric group, we obtained new unexpected results because of our use of a computer algebra system. This was a direct result of being forced to compute in the ring of polynomials modulo the cyclotomic polynomial, after failing to be able to work directly with primitive roots of unity. This text tells the story of how our computer experiments led us to conjecture surprising new identities.