3: Factorization

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Page ID: 82466

Michael Janssen & Melissa Lindsey
Dordt University & University of Wisconsin-Madison

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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.

3.1: Factoring Polynomials
In this section, our first goal will be to extend familiar properties from Z to F[x]. We will also see that particular features of a polynomial (e.g., its degree, or the existence of roots) allows for additional criteria for its irreducibility to be decided.
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.
3.3: Nonunique Factorization
Despite the evidence to the contrary, not every ring has the unique factorization property.