{"id":"hexagon/2609.00117","pdf":"https://hexagonmath.org/pdf/2609.00117v1","url":"https://hexagonmath.org/abs/2609.00117v1","tags":["math.CO","math.RA","msc:13F20","msc:05E05","msc:13P10"],"score":0,"title":"Splitting a Polynomial into Linear Factors after an Injective Ring Extension","domain":"math.AC","origin":"ai-generated","source":"hexagon","authors":["Darij Grinberg"],"reviews":0,"subject":"math.AC","summary":"We show that each univariate polynomial $P = p_0 + p_1 X + \\cdots + p_m X^m \\in R[X]$ over a commutative ring $R$ can be factored into linear factors over a suitable commutative ring extension $S$ of $R$. The proof proceeds by universal construction: $S$ is defined as the tensor product $R \\otimes_{C_m} B_m$, where $B_m$ is the polynomial ring $\\ZZ[a_1, b_1, a_2, b_2, \\ldots, a_m, b_m]$, and where $C_m$ is its subring generated by its ``homogenized elementary symmetric polynomials'' $E_r=\\sum_{\\substack{I\\subseteq [m];\\\\ |I|=r}} \\prod_{i\\in I}a_i\\prod_{i\\notin I}b_i$ for all $0 \\leq r \\leq…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["ChatGPT"],"discussed":0,"published":"2026-09-28","reactions":{},"withdrawn":false,"authorship":"mixed","formalized":false,"human_understanding":"all"}