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Unformatted text preview: extension [ Q ( 6 , 10 , 15): Q ] equals 4 and not 8. 7. (10 points) Let K L be a nite eld extension, and let f be an irreducible polynomial with coecients in K . Assume that the degree of f , and [ L : K ] are relatively prime. Prove that f has no roots in L . 8. (10 points) Let R be a commutative ring with identity and let I and J be ideals of R . Prove: if I + J = R then we also have I 2 + J 3 = R . 9. Let R be a commutative ring and let M be a cyclic R-module, that is, M is generated by a single element. Prove that there exists an ideal I in R such that M = R/I . 1 10. (10 points) Let A 2 2 2 2 2 0 2 0 2 be the presentation matrix for the abelian group X , that is we have the pre-sentation Z 3 A Z 3 X . Find a direct sum of cyclic groups which is isomorphic to X . 2...
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- Spring '11