Goldberg-Coxeter (G-C) transformations generate infinite families of fullerenes while preserving the global shape of their parent structures. A central problem is to determine whether a given fullerene arises from a nontrivial G--C transformation and, if so, to identify and uniquely invert the transformation. In this paper, G-C transformations are formulated within the arithmetic framework of Eisenstein integers. Each transformation corresponds to multiplication by an Eisenstein integer, and its areal scale factor coincides with the associated Eisenstein norm. Since the ring \( \mathbb{Z}[\omega] \) is a unique factorization domain, every G-C transformation admits a canonical decomposition into primary sub-transformations determined by the prime factorization of its norm. We distinguish two types of Eisenstein primes: rational primes corresponding to linear \( k \)-inflations, and non-rational primes governing genuine area-expanding transformations. This arithmetic structure reduces the identification and inversion of G-C fullerenes to prime factorization and divisibility testing in \( \mathbb{Z}[\omega] \), thereby providing a rigorous and practical framework for fullerene classification.