Algebra2007aug - extension [ Q ( 6 , 10 , 15): Q ] equals 4...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 2
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

Page1 / 2

Algebra2007aug - extension [ Q ( 6 , 10 , 15): Q ] equals 4...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online