Algebra Prelim, August 2007

Do all problems

  1. How many elements of order 7 must there be in a simple group of order 168?

  2. Let ρ be a primitive 4th root of 1 over Q.
    Compute the Galois group of (x4 -2)(x2 - 3) over Q and Q(ρ).
    Is Q(ρ) Galois over Q? (Explain your answer.)
    Are there any proper subfields of the splitting field of (x4 -2)(x2 - 3) over Q(ρ) that are Galois over Q(ρ)? (Explain your answer.)

  3. Suppose that 0 --> A -- f--> B -- g--> C --> 0 is a split exact sequence of left R-modules, where R is a ring with a 1. If D is a right R-module, prove that 1 $ \otimes$ f : D $ \otimes_{R}^{}$ A --> D $ \otimes_{R}^{}$ B is a monomorphism.

  4. Let R be a PID and let S be a multiplicatively closed subset of R. Assume that S is nonempty and that S does not contain 0. Prove that S-1R is a PID.

  5. Prove that a group of order 24·112 is solvable.

    1. Prove that f (x) = x4 + 9x - 30 is an irreducible polynomial over Q.
    2. Let g(x) = x2 + 2 and let I be the ideal in Q[x] generated by the product f (x)g(x). Show that Q[x]/I is the product of two fields. What is the dimension over Q of these fields?

  6. Let R be an integral domain. If X is an R-module, then let t(X) denote the subset {xX | rx = 0 for some nonzero rR}.
    1. Prove that t(X) is a submodule of X.
    2. Prove that t(X/t(X)) = 0.
    3. Prove that if X/t(X) is a nonzero cyclic R-module, then X is isomorphic to t(X)⊕R.

Peter Linnell 2007-08-11