Factorization into Irreducibles
DOI link for Factorization into Irreducibles
Factorization into Irreducibles book
Now we turn to the vexed but vital question of uniqueness of factorization in the ring of integers of an algebraic number field. Historically, experience with unique factorization of integers and polynomials over a field led to a general intuition that factorization of algebraic integers should also be unique. In the early days of algebraic number theory many experts, including Euler, simply assumed uniqueness without perceiving any need for a proof, and used it implicitly to ‘prove’ results that were later found to be based on a false assumption.