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.

   (a)

   Compute the Galois group of (x⁴ -2)(x² - 3) over Q and Q(ρ).

   (b)

   Is Q(ρ) Galois over Q? (Explain your answer.)

   (c)

   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.

6. 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?

7. Let R be an integral domain. If X is an R-module, then let t(X) denote the subset {x∈X | rx = 0 for some nonzero r∈R}.
   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.