Algebra Preliminary Exam, Spring 1986

  1. Let F be the free group on the set , and let H be the normal subgroup of F generated by . Prove that F/H is isomorphic to the free abelian group on X.
  2. Let G be an abelian group of order p6, and let . Suppose that |H|=p2. Give all possible such groups G.
  3. Prove that S4 is solvable.
  4. In S5, how many Sylow subgroups of each type are there?
    1. Let R be a PID and let S be a multiplicatively closed subset. Show that S-1R is a PID.
    2. Give an example of a PID with exactly 3 nonassociate irreducible elements.
  5. Let be the set of maximal ideals in a commutative ring R with identity. Set . For , prove that if and only if 1+rs is a unit for all .
  6. Let be a root of x3+4x+2.
    1. Find a basis for over . Justify y our answer.
    2. Express in terms of the basis.
    3. Express in terms of the basis.
  7. Let K be a subfield of the field L, and let such that is odd. Prove that .

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Peter Linnell
Wed Jul 31 14:18:12 EDT 1996