4 New Generalized Criteria for Irreducible Polynomials over Q

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