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Unformatted text preview: 18.781, Fall 2007 Problem Set 10 Due: MONDAY, November 19 A few more problems on algebraic numbers and the general reciprocity program: 1. Show that sin( / 12) is algebraic. 2. In class, we determined that Z [ - 5] is not a principal ideal domain: We stated that unique factorization in a ring R implies R is a principal ideal domain. This means that every ideal in R is generated by a single element. Moreover, Z [ - 5] does not have unique factorization since 6 = 2 3 = (1 + - 5) (1- - 5) . (a) Explain how, using the norm in Z [ - 5], this factorization of 6 im- plies that Z [ - 5] is not a principal ideal domain. (In particular, why cant the factorizations of 6 factor further?) (b) Investigate the ideal (3 , 1 + - 5), the ideal generated by 3 and 1 + - 5 (i.e. the smallest ideal containing these two elements in Z [ - 5]). Can you describe the set of elements in the ideal more concretely? Does the ideal consist of the entire ring Z [ - 5]? Is it a prime ideal? (Recall, an ideala prime ideal?...
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