Maximal Ideals - Algebra - Exam, Exams of Algebra

This is the Exam of Algebra which includes Symmetric Matrix, Eigenvalues, Solvable Group, Eigenvalues, Linear Operator, Simple Group, Nontrivial Group, Irreducible Polynomial etc. Key important points are: Maximal Ideals, Commutative Ring, Minimal Prime Ideal, Rings, Splitting field, Polynomial, Rational Canonical Form, Jordan Canonical Form, Integer Matrix, Ascending Chain

Typology: Exams

2012/2013

Uploaded on 02/21/2013

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Algebra Ph.D. Qualifying Exam, January 2012
Answer all questions. Partial credit will be given.
1. Prove that a free group on any generating set with more than one element is non-Abelian.
2. Prove that every commutative ring with 1 6= 0 has a minimal prime ideal. You may freely use
the fact that such rings have maximal ideals if that is helpful.
3. Compute the Galois group over Qfor the splitting field of the polynomial x4+x3+x2+x+ 1.
4. Find the rational canonical form and Jordan canonical form for the integer matrix
123
456
789
.
5. Prove that Aut(Q8)
=S4where Q8=hi, j, k :i2=j2=k2=1,(1)2= 1, ij =k=
ji, ki =j=ik, jk =i=kjiis the quaternion group. (You are free to use any presenta-
tion of this group you like.)
6. Prove that a PID has the property that any ascending chain of ideals stabilizes.
7. Classify all groups of order p2, where pis prime.
8. Let Fbe a field. Let Vbe a finite dimensional vector space. Prove that Vis isomorphic to its
double dual (V).
9. Is the (possibly infinite) direct product of fields a PID? Justify your answer.
10. Let pbe a prime. Prove that there is a field of order pn, for each n1.
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Algebra Ph.D. Qualifying Exam, January 2012 Answer all questions. Partial credit will be given.

  1. Prove that a free group on any generating set with more than one element is non-Abelian.
  2. Prove that every commutative ring with 1the fact that such rings have maximal ideals if that is helpful. 6 = 0 has a minimal prime ideal. You may freely use
  3. Compute the Galois group over Q for the splitting field of the polynomial x^4 + x^3 + x^2 + x + 1.
  4. Find the rational canonical form and Jordan canonical form for the integer matrix

^147 258

  1. Prove that Aut( −ji, ki = j = −Qik, jk 8 ) ∼= =S 4 i where= −kj (^) 〉Q is the quaternion group. (You are free to use any presenta- 8 = 〈i, j, k : i^2 = j^2 = k^2 = − 1 , (−1)^2 = 1, ij = k = tion of this group you like.)
  2. Prove that a PID has the property that any ascending chain of ideals stabilizes.
  3. Classify all groups of order p^2 , where p is prime.
  4. Letdouble dual ( F be a field. LetV ∗)∗. V be a finite dimensional vector space. Prove that V is isomorphic to its
  5. Is the (possibly infinite) direct product of fields a PID? Justify your answer.
  6. Let p be a prime. Prove that there is a field of order pn, for each n ≥ 1.

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