Unique Factorization

Abstract

In this chapter we show that any polynomial of degree ≥ 1 with coefficients in a field factors uniquely (in a sense to be defined) into a product of irreducible polynomials. To reach this result, we follow the same development as for natural numbers: the division theorem, Euclid’s algorithm, Bezout’s identity. But just the first part of this development is enough to complete a proof that for any prime number p, there is a number b so that every number prime to p is congruent modulo p to a power of b. This result, the primitive root theorem, has very interesting consequences, as we’ll see starting in Chapter 23.