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Unformatted text preview: S I = /0. Prove that S1 I : = { x / s  x I and s S } R and S1 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 comprehensive (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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 Spring '08
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 Math, Algebra

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