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College Algebra Exam Review 286

# College Algebra Exam Review 286 - Proof Exercises 6.5.1...

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296 6. RINGS (b) For all d , if d divides each a i , then d divides c . Elements a i ; : : : ; a s in an integral domain are said to be relatively prime if 1 is a gcd of f a i g . Given any Euclidean domain R , we can follow the proofs given for the integers and the polynomials over a field to establish the following results: Theorem 6.5.8. Let R be a Euclidean domain. (a) Two nonzero elements f and g 2 R have a greatest common divisor which is contained in the ideal Rf C Rg . The greatest common divisor is unique up to multiplication by a unit. (b) Two elements f and g 2 R are relatively prime if, and only if, 1 2 Rf C Rg . (c) Every ideal in R is principal. (d) Every irreducible element is prime (e) Every nonzero, nonunit element has a factorization into irre- ducibles, which is unique (up to units and up to order of the factors).
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Unformatted text preview: Proof. Exercises 6.5.1 through 6.5.3 . n This result suggests making the following deﬁnitions: Deﬁnition 6.5.9. An integral domain R is a principal ideal domain (PID) if every ideal of R is principal. Deﬁnition 6.5.10. An integral domain is a unique factorization domain (UFD) if every nonzero element has a factorization by irreducibles that is unique up to order and multiplication by units. According to Theorem 6.5.8 , every Euclidean domain is both a prin-cipal ideal domain and a unique factorization domain. The rest of this section will be devoted to a further study of principal ideal domains and unique factorization domains; the main result will be that every principal...
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