From Polynomial Ring to α-ideal Topological Space

Abstract

    The reduction of a polynomial (poly) over Galois Fields ( ) reveals whether the coefficients of that poly truly belong to , based on their irreducibility into non-trivial polys.  is constructed from the primitive poly  through the ring . Furthermore, this study is related to the framework of ideal topological spaces, where a binary relation is derived through the algebraic structure represented by the -adic valuation function . Starting from the equivalence classes derived from the valuation based relation, a partition topology is generated on the finite field. Combining the ideal I with the topological structure enables us to define both the closure operator and the -local function. The significance of these two operators becomes evident through their ability to classify field elements with high precision, there by revealing the deep structure of finite fields. Furthermore, the -ideal topology provides a more refined partition of the field elements, the filtration of negligible elements through the ideal, and a systematic study of the structural properties of polynomials.