math.AC — Commutative Algebra

Splitting a Polynomial into Linear Factors after an Injective Ring Extension

Contributed by Darij Grinberg

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 $0 \leq r \leq m$. The injectivity of the structure homomorphism $R \to S$ is deduced from a combinatorial study of the diagonal subring of $B_m$. In the process, a homogeneous variant of the Garsia--Stanton basis is constructed, and some classical properties of symmetric polynomials are recovered.