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College Algebra Exam Review 291

College Algebra Exam Review 291 - 6.5 FACTORIZATION(a(b 301...

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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 OE p be the subring of C generated by Z and p 2 . Show that Z OE p D f a C b p 2 W a; b 2 Z g . Show that Z OE 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 OE!Ł be the subring of C generated by Z and ! . Show that Z OE!Ł D f a C b! W a; b 2 Z g . Show that Z OE!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 OEiŁ of 14 C 2i and 21 C 26i . 6.5.7. Compute a greatest common divisor in Z OEiŁ of 33 C 19i and 18 16i . 6.5.8. Show that for two elements a; b in an integral domain R , the fol- lowing are equivalent: (a) a divides b and
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