3: Factorization
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 \(F[x]\). 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 1.2.4 and Theorem 2.4.9 , 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.
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- 3.2: Factorization in Euclidean Domains
- In this section, our explorations of the structural arithmetic properties that guarantee unique factorization culminate in Theorem 3.2.7 . Specifically, we'll see that all Euclidean domains possess the unique factorization property. To prove this theorem, we will rely in part on an interesting property of chains of ideals in Euclidean domains.