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Unformatted text preview: Algebra , Assignment 11 SOLUTIONS 1. Call R a TurkeyYam Ring if it is an Integral Domain and so that it has a “size function” d : R −{ } → { , 1 , 2 ,... } with d ( α ) ≤ d ( αβ ) and with the following property: For all α,β ∈ R −{ } with α not dividing β there exist m,n ∈ R with mα + nβ negationslash = 0 and d ( mα + nβ ) < d ( α ). Prove that a TurkeyYam Ring is a Principle Ideal Domain. Solution: Let I be an ideal of R . If I = { } then I = (0) and we are done. Otherwise, let α be a nonzero element of I with minimal d ( α ). We claim I = ( α ). Otherwise, let β ∈ I , β negationslash∈ ( α ). Let m,n be as above. Then mα + nβ ∈ I with a smaller size than α , a contradiction. 2. Give the Prime Factorization in Z [ i ] of: 6 , 10 + 3 i, 4 + 7 i . Solution: 6 − 3 × 2 = ( − i ) · 3 · (1 + i ) 2 10 + 3 i is prime as d (10 + 3 i ) = 109 is a prime in Z 4 + 7 i = (2 + i ) · (3 + 2 i ) and both are primes as they have sizes 5 , 13, both primes in Z . 3. (*) Let a,b be relatively prime and nonzero....
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 Fall '08
 JOSEPHS
 Algebra, Prime number, Greatest common divisor, Euclidean algorithm, Principal ideal domain

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