4 New Generalized Criteria for Irreducible Polynomials over Q

This paper presents a unified and highly optimized algebraic framework for determining polynomial irreducibility over the rational field Q, general Dedekind domains, and multivariate polynomial rings. While geometric approaches like Newton
polygons and p-adic valuations are historically profound, applying them manually
or computationally often involves cumbersome graphical constructions or expensive factorization algorithms. We bridge this gap by introducing four generalized,
explicit criteria utilizing polynomial translation constants and arithmetic progressions (both ascending and descending) within the p-adic valuations of coefficients.
Although rigorously grounded in established foundational theorems—specifically
Dumas’s Irreducibility Theorem (1906), Mac Lane’s residual polynomials (1936),
and the geometric theory of Newton polytopes—these explicit algebraic formulations are novel. They successfully package complex geometric constraints into
simple, highly memorable, and fast-to-verify arithmetic inequalities. Ultimately,
this research provides a uniquely convenient and immensely practical toolkit that
streamlines irreducibility testing, bypassing the need for geometric plotting and
offering rapid algebraic certificates for computational number theory and cryptographic parameter generation.