An alternative perspective on recursive factorization of $ x^n-1 $ over finite fields and applications in coding theory

Building on the well-known factorization of $ x^n-1 $ over finite fields $ \mathbb{F}_q $, this paper presents an alternative perspective on its irreducible factors in arithmetic settings where they admit explicit recursive descriptions. As applications, these results are used in the characterization and enumeration of self-orthogonal cyclic codes over finite fields. We first consider the case $ n = p^s $, where $ p $ is an odd prime coprime to $ q $, and show that suitable irreducible factors can be obtained recursively by power substitutions from factors at a critical degree. This recursive viewpoint is then extended to a product of two odd prime powers and, more generally, to certain finite products of distinct odd prime powers under suitable separation conditions on multiplicative orders. From these recursive descriptions, we derive explicit formulas for the numbers of monic irreducible factors of $ x^n-1 $ over $ \mathbb{F}_q $. We then apply the same framework to the study of self-reciprocal irreducible monic and self-conjugate-reciprocal irreducible monic factors of $ x^n-1 $. In particular, we obtain structural descriptions in extremal cases where all irreducible factors are of the corresponding type or where $ x-1 $ is the unique such factor. These results are further applied to the enumeration of Euclidean and Hermitian self-orthogonal cyclic codes. In this way, the paper provides an explicit recursive and enumerative framework connecting the factorization patterns of $ x^n-1 $ with counting problems and constructions in algebraic coding theory.