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Splitting a Polynomial into Linear Factors after an Injective Ring Extension

A mix of human-written and AI-generated textHuman understanding: all partsmath.AC — Commutative Algebramath.CO — Combinatoricsmath.RA — Rings and Algebras

Contributed by Darij Grinberg ↗

SubmitterDarij Grinberg

Version 1 / Sep 28, 2026 / CC0 1.0

Abstract

We show that each univariate polynomial P=p0+p1X+⋯+pmXm∈R[X]P = p_0 + p_1 X + \cdots + p_m X^m \in R[X] over a commutative ring RR can be factored into linear factors over a suitable commutative ring extension SS of RR. The proof proceeds by universal construction: SS is defined as the tensor product R⊗CmBmR \otimes_{C_m} B_m, where BmB_m is the polynomial ring $\ZZ[a_1, b_1, a_2, b_2, \ldots, a_m, b_m]$, and where CmC_m is its subring generated by its ``homogenized elementary symmetric polynomials'' Er=∑I⊆[m];∣I∣=r∏i∈Iai∏i∉IbiE_r=\sum_{\substack{I\subseteq [m];\\ |I|=r}} \prod_{i\in I}a_i\prod_{i\notin I}b_i for all 0≤r≤m0 \leq r \leq m. The injectivity of the structure homomorphism R→SR \to S is deduced from a combinatorial study of the diagonal subring of BmB_m. In the process, a homogeneous variant of the Garsia--Stanton basis is constructed, and some classical properties of symmetric polynomials are recovered.

Provenance statement

Generated by GPT-5.5 over a long conversation, with significant steering and rewriting. Also available at https://www.cip.ifi.lmu.de/~grinberg/algebra/factorpoly-gpt.pdf and https://www.cip.ifi.lmu.de/~grinberg/algebra/factorpoly-gpt.tex

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Version history

  1. v1Initial depositCurrentSep 28, 2026