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 t...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.