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November 25th, 2006, 05:41 PM  #1 
Newbie Joined: Nov 2006 From: USA Posts: 7 Thanks: 0  Unique factorization Domain
Let R be the quadratic integer ring Z[sqrt(5)] and define the ideals I_2=(2, 1 + sqrt(5)); I_3=(3, 2 + sqrt(5)); I'_3=(3, 2  sqrt(5)); a) Prove that I_2, I_3, I'_3 are prime ideals in R. b) Show that the factorizations 6=2.3=(1+sqrt(5))(1sqrt(5)) imply the equality of ideals (6)=(2)(3) and (6)=(1+sqrt(5))(1sqrt(5)). Show that these two ideal factorizations give the same factorization of the ideal (6) as the product of prime ideals. 

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domain, factorization, unique 
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