4 New Generalized Criteria for Irreducible Polynomials over Q
Description
This paper presents a comprehensive and highly generalized geometric framework for determining polynomial irreducibility over the rational field Q, global fields, and multivariate polynomial rings. While the classical Eisenstein criterion remains a foundational tool in abstract algebra, its strict divisibility requirements severely limit its applicability. We construct a robust theoretical extension by integrating polynomial translation constants and arithmetic progressions within the p-adic valuations of coefficients, specifically utilizing both positive and negative shifts (p m±ik). Furthermore, we systematically elevate this framework across multiple dimensions of algebraic abstraction. First, we relax the strict coprimality constraint by employing Mac Lane’s residual polynomials over finite fields for cases where the greatest common divisor d > 1. Second, we abstract the criteria from the rational integers to general Dedekind domains using p-adic valuations. Third, we introduce a non-linear bounding mechanism that accommodates arbitrary strictly descending or ascending valuation sequences. Finally, we transcend two-dimensional Newton polygons by applying (r + 1)-dimensional Newton polytopes and bounding hyperplanes to evaluate multivariate polynomials. Ultimately, this research provides a unified, elementary, yet immensely powerful toolkit that streamlines irreducibility testing in computational number theory, algebraic geometry, and post-quantum 1 cryptographic parameter generation. Keywords: Irreducible polynomials, Generalized Eisenstein criterio