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College Algebra Exam Review 285

College Algebra Exam Review 285 - Definition 6.5.5 A...

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6.5. FACTORIZATION 295 trick to establishing the Euclidean property is to work temporarily in the field of fractions, the set of complex numbers with rational coefficients. Let z; w be nonzero elements in Z OEiŁ . There is at least one point q 2 Z OEiŁ whose distance to the complex number z=w is minimal; q satisfies j< .q z=w/ j 1=2 and j= .q z=w/ j 1=2 . Write z D qw C r . Since z and qw 2 Z OEiŁ , it follows that r 2 Z OEiŁ . But r D .z qw/ D .z=w q/w , so d.r/ D j z=w q j 2 d.w/ .1=2/d.w/ . This completes the proof that the ring of Gaussian integers is Euclidean. Let us introduce several definitions related to divisibility before we continue the discussion: Definition 6.5.3. Let a be a nonzero, nonunit element of an integral do- main. A proper factorization of a is an equality a D bc , where neither b nor c is a unit. The elements b and c are said to be proper factors of a . Definition 6.5.4. A nonzero, nonunit element of an integral domain is said to be irreducible if it has no proper factorizations.
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Unformatted text preview: Definition 6.5.5. A nonzero nonunit element a in an integral domain is said to be prime if whenever a divides a product bc , then a divides one of the elements b or c . An easy induction shows that whenever a prime element a divides a product of several elements b 1 b 2 ³³³ b s , then a divides one of the elements b i . In any integral domain, every prime element is irreducible (Exercise 6.5.19 ), but irreducible elements are not always prime. Definition 6.5.6. Two elements in an integral domain are said to be asso-ciates if each divides the other. In this case, each of the elements is equal to a unit times the other. Definition 6.5.7. A greatest common divisor (gcd) of several elements a i ;:::;a s in an integral domain R is an element c such that (a) c divides each a i , and...
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