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Chapter 3 Factorization
In this chapter, we come to the heart of the text: a structural investigation of unique factorization in the familiar contexts of
\(\Z\) 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
\(\Z\) 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.
