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November 25th, 2006, 05:41 PM   #1
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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))(1-sqrt(-5))
imply the equality of ideals (6)=(2)(3) and (6)=(1+sqrt(-5))(1-sqrt(-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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