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.

Abstract Algebra (5): Rings and Ideals — When Multiplication Enters the Picture

Thu, 09 Sep 2021 09:00:00 +0000

Groups capture symmetry through a single operation. But most of the number systems we actually compute with — integers, polynomials, matrices — carry two operations that interact: addition and multiplication. The moment you want to talk about divisibility, factorization, or solving equations, one operation is not enough. You need a ring.

This article develops ring theory from scratch: the axioms, the key examples, the pathologies that make ring theory richer (and harder) than group theory, and the central concept of an ideal — the ring-theoretic analogue of a normal subgroup. By the end you will have the language to state the First Isomorphism Theorem for rings and to understand why “modding out by an ideal” is the right way to build new rings from old ones.

Ring-Theory on Chen Kai Blog

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

Abstract Algebra (5): Rings and Ideals — When Multiplication Enters the Picture