Polynomial and Euclidean Rings
William J. Gilbert, W. K. Nicholson · 2003
Chapter 9 investigates rings of polynomials over a field, and their generalization called euclidean rings, that possess a division algorithm. These euclidean rings are shown to have a euclidean algorithm, and the unique factorization property. One important euclidean ring is the ring of gaussian integers. The question of whether a polynomial is reducible or not is crucial in Chapter 10, so various methods are given for factoring polynomials over the complex numbers, the rational numbers, and over finite fields. The chapter ends with the Chinese Remainder Theorem, that can be used to represent numbers as residues of prime powers.