New Irreducibility Criteria for Polynomials over Q via Prime Progressions and Coprime Base

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