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.
KeywordsCommutative Ring Unique Factorization Primitive Element Great Common Divisor Irreducible Polynomial
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