Rings with Inquiry: An inquiry-oriented introduction to ring theory.

In this chapter, we come to the heart of the text: a structural investigation of unique factorization in the familiar contexts of $\mathbb{Z}$ and $F[x]\text{.}$ In Section 3.1, we explore theorems that formalize much of our understanding of that quintessential high school algebra problem: factoring polynomials. As we saw in Theorem 1.2.5 and Theorem 2.4.14, both $\mathbb{Z}$ and $F[x]$ have a division algorithm and, thus, are Euclidean domains. In Section 3.2, we explore the implications for multiplication in Euclidean domains. That is: given that we have a well-behaved division algorithm in an integral domain, what can we say about the factorization properties of the domain?

Finally, in the optional Section 3.3, we explore contexts in which unique factorization into products of irreducibles need not hold.