This paper examines the reduction of polynomials over Galois Fields (GFs) to determine whether their coefficients truly belong to the field, based on irreducibility into non-trivial polynomials. The field GF(2^n) is constructed as a quotient ring formed from a primitive irreducible polynomial of degree n over the binary polynomial ring, providing an ideal algebraic environment for studying finite field structures. The study links this algebraic construction to the theory of Approximation Spaces, where a binary relation generated through the algebraic conjugacy relation is proven to be an equivalence relation, and is used to derive the two approximation limits (lower and upper) of subsets within the field. Through these relations, the polynomials are partitioned into equivalence classes, which in turn serve as the basis for a topological space, allowing the study of proximity and connectivity among field elements. This approach enables a classification of subsets into exact and rough sets, offering a deeper understanding of the structure of GFs and their irreducible polynomials from both an algebraic and topological perspective. For precise partitioning and classification of field elements, an accuracy (rough membership) measure is introduced, quantifying the degree of belonging of elements to a subset and distinguishing exact sets from rough ones. This combined algebraic-topological-rough set framework offers a systematic approach to analyzing polynomial characteristics within approximation spaces and is shown to have promising applications in cryptographic frameworks, where precise.