New Irreducibility Criteria for Polynomials over Q via Prime Progressions and Coprime Base
Description
This paper introduces four novel deterministic criteria for establishing the irre ducibility of polynomials over the rational field Q. While classical methods often rely on local p-adic properties of a single prime (e.g., Eisenstein’s criterion), our approach focuses on the global architectural patterns of the entire coefficient set. By integrating geometric root-bounding techniques via the Enestr¨om-Kakeya theo rem with prime number theory, we prove that polynomials whose coefficients form strictly monotonic sequences of consecutive primes or arithmetic progressions of primes are unconditionally irreducible. Furthermore, we extend this framework to multiplicative structures using pairwise coprime bases with arithmetic progressions in their exponents. We explicitly distinguish our results from the classical tradi tions of Cohn, Murty, and Guersenzvaig by shifting the focus from specific base representations to global structural monotonicity. These criteria offer O(n)-time complexity sieves, bypassing the need for computationally intensive factorization algorithms