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Unique Factorization

  • Lindsay N. Childs
Part of the Undergraduate Texts in Mathematics book series (UTM)

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.

Keywords

Commutative Ring Unique Factorization Primitive Element Great Common Divisor Irreducible Polynomial 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 1995

Authors and Affiliations

  • Lindsay N. Childs
    • 1
  1. 1.Department of MathematicsSUNY at AlbanyAlbanyUSA

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