text: # Algebra Qualifying Examination January 7, 2008 Instructions:
  
  ```Algebra Qualifying Examination
  January 7, 2008
  Instructions:
  • There are eight Algebra questions worth a total of 100 points. Individual point
  values are listed next to each problem.
  as well as the clarity and main steps of your reasoning. “Rough working” will
  not be accepted: answers must be written in a structured and understandable
  manner.
  • You may use a calculator to check your computations (but you will not earn
  points for using it as a step in your reasoning).
  Notation: Throughout, Z denotes the integers, Q denotes the rational numbers, R
  denotes the real numbers, and C denotes the complex numbers.
  1. (15 points) Let n be a positive integer. Denote by Sn and An the symmetric
  and alternating groups of degree n, respectively.
  (a) For each positive integer n, give a composition series of Sn .
  (b) Find the centralizer in S5 of the 3-cycle (345).
  (c) Find the centralizer in A5 of the 3-cycle (345).
  2. (15 points) List all possible groups of order 175 (up to isomorphism).
  3. (10 points) Let R be a ring, S a subring of R, and I an ideal of R. Suppose
  that R = S + I and S ∩ I = 0.
  (a) Prove that each element of R may be written r = s + i for unique elements
  s ∈ S and i ∈ I.
  (b) Prove that R/I is isomorphic to S by giving a ring isomorphism from R/I
  to S.
  4. (10 points) Let R1 and R2 be integral domains with fields of fractions (i.e.
  quotient rings) F1 and F2 , respectively. Suppose that there is an injective ring
  homomorphism φ : R1 → R2 . Prove that φ extends to F1 , i.e. there is an
  injective ring homomorphism ψ : F1 → F2 such that ψ(r) = φ(r) for all r ∈ R.
  5. (10 points) Let R be a ring. Let P1 and P2 be R-modules. Prove that P1 ⊕ P2
  is a projective R-module if, and only if, both P1 and P2 are projective.
  6. (10 points) Let A be a finitely generated abelian group.
  (a) If A is finite, prove that A ⊗Z Q = 0.
  (b) If A is infinite, prove that A ⊗Z Q ∼
  = Qr , an isomorphism of Z-modules,
  for some positive integer r.
  7. (15 points)
  √
  √
  (a) Q
  Express the minimal polynomial f (x) of 2 + 3 over Q as a product
  (x − αi ), for complex numbers αi .
  √
  √
  (b) Q
  Express the minimal polynomial g(x) of 3 2 + 3 4 over Q as a product
  (x − βi ), for complex numbers βi .
  (c) What is the Galois group of f (x)g(x) over Q?
  √ √
  8. (15 points) Let
  W
  =
  Q(
  2, −2).
  √ √
  (a) Extend { 2, −2} to a basis of W as a vector
  √ space over Q.
  (b) Let f : W → W be defined by f (α) = (1 + 2)α. Verify that f is a linear
  transformation of W as a vector space over Q and find its characteristic
  polynomial.
  (c) Find a basis of eigenvectors for f of C ⊗Q W .
  ```
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