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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 ﬁnite ﬁeld extension, and let f be an irreducible polynomial with coeﬃcients 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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This note was uploaded on 06/19/2011 for the course MATH 699 taught by Professor Na during the Spring '11 term at University of North Carolina School of the Arts.
- Spring '11