# Algebra2007aug - extension Q √ 6 √ 10 √ 15 Q equals 4...

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Ph. D. Algebra Preliminary Exam Monday, August 20, 2007 Show the work you do to obtain an answer. Give reasons for your answers. There are 10 questions. You should answer each question. 1. (10 points) Let G be a group of order p 2 for some prime number p . Prove that G is abelian. 2. (10 points) Prove that a ﬁnite group of order 24 is not simple. 3. (10 points) Let H K G be groups. Prove that H is normal in K if and only if K N G ( H ) where N G ( H ) is the normalizer of H in G . 4. (10 points) Let T : C n C n be a linear operator. Prove that ker T = (im T * ) , where the orthogonal complement is taken with respect to the usual hermition inner product on C n . 5. (10 points) Let A be an n × n real matrix with transpose A T , and prove that the following (criteria for A to be orthogonal) are equivalent. 1. AX = X for all X R n , where ∥ · ∥ is the usual norm on R n ; 2. AX,AY = X,Y for all X,Y R n , where ⟨· , · , is the usual inner product on R n ; 3. A T A = I n , the n × n identity matrix. 6. (10 points) Let Q be the rational numbers. Prove the degree of the ﬁeld

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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.

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Algebra2007aug - extension Q √ 6 √ 10 √ 15 Q equals 4...

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