homework10 - Math 113 Section 5 Fall 2010 Homework#10 Due...

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Math 113 Section 5 Homework #10 Fall 2010 Due: Monday, November 22 1. Section 8.1, Exercise 3: Let R be a Euclidean domain. Let m be the minimum integer if the set of nonzero elements of R (i.e., let m = min { N ( r ) | r R - { 0 }} ). Prove that every nonzero element of R of norm m is a unit. Deduce that a nonzero element of norm zero (if such an element exists) is a unit. 2. Section 8.1, Exercise 7: Find a generator for the ideal (85 , 1+13 i ) in Z [ i ] , i.e., a greatest common divisor for 85 and 1 + 13 i , by the Euclidean Algorithm. Do the same for the ideal (47 - 13 i, 53 + 56 i ) . 3. Section 8.2, Exercise 3: Prove that the quotient of a principal ideal domain by a prime ideal is again a principal ideal domain. 4. Section 8.2, Exercise 5: Let R be the quadratic integer ring Z [ - 5] . Define the ideals I 2 = (2 , 1 + - 5) , I 3 = (3 , 2 + - 5) , and I 3 = (3 , 2 - - 5) . (a) Prove that I 2 , I 3 and I 3 are nonprincipal ideals in R . (b) Prove that the product of two nonprincipal ideals can be principal by showing that I 2 2 is the principal ideal generated by 2 , i.e, I 2 2 = (2) .
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