sf - S I = /0. Prove that S-1 I : = { x / s | x I and s S }...

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Math 4124 Wednesday, April 27 Sample Final Problems since Second Test 1. Determine the number of isomorphism classes of noncyclic abelian groups of order 3200. 2. Determine which of the following define an ideal I of Z [ x ] . For those which are ideals, describe I 2 . (a) All polynomials whose constant coefficient is a multiple of 2 and whose coeffi- cient of x is a multiple of 4. (b) All polynomials whose constant coefficient is a multiple of 4 and whose coeffi- cient of x is a multiple of 2. 3. Let R be a commutative ring and let I denote the set of nilpotent elements of R (that is, { r R | r n = 0 for some positive integer n } ). Prove that I ± R and that R / I has no nonzero nilpotent elements. 4. Prove that the following rings are not isomorphic. (a) Z / 4 Z and Z / 2 Z × Z / 2 Z . (b) Q [ 2 ] and Q [ 3 ] , where for n Z , Q [ n ] = { a + b n | a , b Q } . 5. Prove that Q [ x ] / ( 2 x - 1 ) = Q . 6. Let R be an integral domain. Suppose 0 6 = x , y R , xR yR , and xR is a prime ideal of R . Prove that yR = xR or R . 7. Let R be an integral domain, let I ± R , and let S be a multiplicatively closed subset of R which contains 1. Suppose
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Unformatted text preview: S I = /0. Prove that S-1 I : = { x / s | x I and s S } R and S-1 I 6 = R . 8. Prove that C [ x ] / ( x 2 + 1 ) = C C . 9. Let R be a PID and let S be an integral domain. If : R S is an epimorphism, prove that either is an isomorphism, or that S is a eld. 10. Let k = Z / 2 Z , the eld with two elements. (a) Prove that x 2 + x + 1 is the only irreducible polynomial of degree 2 in k [ x ] . (b) Prove that k [ x ] / ( x 4 + x + 1 ) is a eld with 16 elements. The exam is on Wednesday May 11, 10:05 a.m. to 12:05 p.m. in Smyth 331. It is com-prehensive (includes material from the rst two tests). The material since the second test is Sections 5.1,5.2, 7.16, and parts of sections 8.2,9.1,9.2. One of the problems will be identical to one of the ungraded homework problems since the second test....
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