Abstract Algebra (6): Polynomial Rings — Factorization and Unique Decomposition

Sat, 11 Sep 2021 09:00:00 +0000

Polynomials are the laboratory of algebra. Nearly every concept in ring theory — ideals, quotients, factorization, irreducibility — was first understood through polynomial examples before being abstracted. This is no coincidence: polynomial rings are rich enough to exhibit all the interesting phenomena yet structured enough to permit explicit computation.

In this article we study the ring $$R[x]$$ of polynomials over a ring $$R$$ . We develop the division algorithm, establish irreducibility criteria (Eisenstein, reduction mod $$p$$ , the rational root test), define Unique Factorization Domains, and prove Gauss’s Lemma — the bridge between factorization over $\mathbb{Z}$ and factorization over $\mathbb{Q}$ . The payoff is a clear picture of when and why unique factorization holds, and what goes wrong when it fails.

Polynomials on Chen Kai Blog

Abstract Algebra (6): Polynomial Rings — Factorization and Unique Decomposition