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# asf - Math 4124 Monday May 2 Solutions to Sample Final...

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Math 4124 Monday, May 2 Solutions to Sample Final Problems 1. 3200 = 2 7 · 5 2 . The number of isomorphism classes of abelian groups of order 5 2 is the number of partitions of 2. The partitions of 2 are 2 and (1,1), hence the number of isomorphism classes of abelian groups of order 5 2 is 2. The number of isomorphism classes of abelian groups of order 2 7 is the number of partitions of 7. The parti- tions of 7 are 7, (6,1), (5,2), (4,3), (5,1,1), (4,2,1), (3,3,1), (3,2,2), (4,1,1,1), (3,2,1,1), (2,2,2,1), (3,1,1,1,1), (2,2,1,1,1), (2,1,1,1,1,1), (1,1,1,1,1,1,1), which gives a total of 15 partitions. Therefore the number of isomorphism classes of abelian groups of order 3200 is 2 · 15 = 30. Since exactly one of these groups is cyclic, we conclude that the number of noncyclic abelian groups is 29. 2. (a) This is not an ideal. For example 2 is in the described subset, but 2 x is not. (b) This is an ideal. Let I be described set. Then I is certainly an abelian group under addition. Finally consider 4 a 0 + 2 a 1 x + a 2 x 2 + ··· + a n x n , where a i Z for all i ; this is the general element of I . The general element of Z [ x ] is b 0 + b 1 x + ··· + b m x m , where b i Z . When we multiply these two elements together, we get 4 a 0 b 0 + 2 ( a 1 b 0 + 2 a 0 b 1 ) x +( 4 a 0 b 2 + 2 a 1 b 1 + a 2 b 0 ) x 2 + ··· + x n + m . The above element is in I and it follows that I is an ideal of Z [ x ] . Finally let 4 c 0 + 2 c 1 x + c 2 x 2 + ··· + c k x k be another element of I . Then I 2 consists of sums of elements of the form 16 a 0 c 0 + 8 ( a 1 c 0 + a 0 c 1 ) x + 4 ( a 2 c 0 + a 1 c 1 + a 0 c 2 ) x 2 + 2 ( 2 a 3 c 0 + a 2 c 1 + a 1 c 2 + 2 a 0 c 3 ) x 3 + ··· + a n c k x n + k . It follows that I 2 consists of all elements whose constant coefficient is a multiple of 16, whose coefficient of x is a multiple of 8, whose coefficient of x 2 is a multiple of 4, and whose coefficient of x 3 is a multiple of 2. The coefficients of x 4 and higher degree terms are arbitrary.

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asf - Math 4124 Monday May 2 Solutions to Sample Final...

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