Factorization Into Primes

Abstract

Traditionally, the theory of hcf’s and lcm’s is developed—if so imposing a phrase may appropriately be applied—from the technique of factorizing whole numbers as a product of prime numbers. We have given, in the preceding chapter, a treatment which is quite independent of such factorization, and we hope the reader will have satisfied himself that the Euclidean algorithm, Theorem 11.5.4 and formula 11.5.5 give a clearer insight into the nature of hcf’s than is provided by factorization into primes. However, it is, of course, a matter of profound importance that certain numbers are prime, that all whole numbers may be expressed as a product of prime numbers, and that such expressions are essentially unique. We proceed in this chapter to give precise meanings to these notions, and to prove them in a Euclidean domain. The final section of the chapter is devoted to transferring the theory from the Euclidean domain ℚ[x] to the non-Euclidean domain ℤ[x]—a delicate operation often neglected in elementary and traditional treatments.