College Algebra Exam Review 291

College Algebra Exam Review 291 - 6.5. FACTORIZATION (a)...

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6.5. FACTORIZATION 301 (a) Every ideal in R is principal. (b) If an irreducible p divides the product of two nonzero elements, then it divides one or the other of them. 6.5.3. Let R be a Euclidean domain. Show that every nonzero, nonunit element has a factorization into irreducibles, which is unique (up to units and up to order of the factors). 6.5.4. Let Z Œ p ± be the subring of C generated by Z and p ± 2 . Show that Z Œ p ± D f a C b p ± 2 W a;b 2 Z g . Show that Z Œ p ± is a Euclidean domain. Hint: Try N.z/ D j z j 2 for the Euclidean function. 6.5.5. Let ! D exp .2±i=3/ . Then ! satisfies ! 2 C ! C 1 D 0 . Let Z Œ!Ł be the subring of C generated by Z and ! . Show that Z Œ!Ł D f a C b! W a;b 2 Z g . Show that Z Œ!Ł is a Euclidean domain. Hint: Try N.z/ D j z j 2 for the Euclidean function. 6.5.6. Compute a greatest common divisor in Z ŒiŁ of 14 C 2i and 21 C 26i . 6.5.7. Compute a greatest common divisor in Z ŒiŁ of 33 C 19i and 18 ± 16i . 6.5.8.
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.

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