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.

本文提出了一个统一且高度优化的代数框架，用于判定有理数域（Q）、一般戴德金整环（Dedekind domains）以及多元多项式环（multivariate polynomial rings）上的多项式不可约性。尽管诸如牛顿多边形（Newton polygons）与p进赋值（p-adic valuations）等几何方法在学术史上颇具影响力，但手动或借助计算手段应用它们时，往往需要繁琐的图形构建或高成本的因式分解算法。我们通过引入四类广义显式判别准则填补了这一空白，这些准则利用系数p进赋值中的多项式平移常数与升、降两类等差数列。尽管这些准则严格基于已有的基础定理——具体为迪马不可约性定理（Dumas’s Irreducibility Theorem，1906）、麦克莱恩剩余多项式（Mac Lane’s residual polynomials，1936）以及牛顿多面体（Newton polytopes）几何理论——但这类显式代数表述仍属首创。它们成功将复杂的几何约束封装为简洁易记、可快速验证的算术不等式。最终，本研究提供了一套极具便捷性与实用性的工具集，可简化不可约性测试流程，无需进行几何绘图，并可为计算数论与密码学参数生成提供快速代数验证凭证。