Chapter 5

Factorization in Integral Domains

There still remain three studies suitable for free man. Arithmetic is one of them.

—Plato

We see therefore that ideal prime factors reveal the essence of complex numbers, make them transparent, as it were, and disclose their inner crystalline structure.

—Ernst Eduard Kummer

We have proved two unique factorization theorems: Every integer greater than one is uniquely a product of primes, and if F is a field every polynomial of positive degree is uniquely a product of an element of F times a product of monic irreducible polynomials. In this chapter, we characterize the integral domains for which a similar theorem holds (called unique factorization domains, or UFDs) and discuss some important classes of UFDs.^{63}

This theory has a long history and can be regarded as one of the original sources of modern abstract algebra. At the beginning of the nineteenth century, Gauss used the fact that the ring (now called the gaussian integers) is a UFD to prove his law of biquadratic reciprocity, a method of determining when the congruence x^{4} ≡ b (mod n) has a solution. Inspired by the fact that i is a (fourth) root of unity, Kummer tried to extend Gauss' work by considering , where is any complex root of unity. However, he discovered that may not be ...