Masters Comprehensive Exam Modern Algebra August 1995

There are 6 questions in all. Be sure to give complete proofs and explanations.

1. (a) If σ is any permutation in the symmetric group S_{n} and (x_{1}, …, x_{r}) is an r-cycle in S_{n}, show that σ(x_{1}, …, x_{r})σ^{−1} is again an r–cycle. (Determine the resulting r–cycle.)

(b) Are any two r–cycles in S_{n} conjugate? If so, prove it. If not, give a counterexample.

2. Let p and q be primes with p < q.

(a) Determine all abelian groups pf order pq^{2}.

(b) Is a group of order pq^{2} always abelian? If so, prove it. If not, give an example of primes p < q and a non-abelain group of order pq^{2}.

3. Let V be a vector space over a field F. Let P be a linear transformation P : V → V such that P^{2} = P, that is, a projection.

(a) Show that V = ℑP ⊕ ker P.

(b) If V is finite dimensional and P and Q are two commuting projections, so P Q = QP, show that V has a basis with respect to which both P and Q are represented by diagonal matrices.

4. Let R be a principal ideal domain (PID) and M a module over R.

(a) Define what it means for M to be a torsion module.

(b) Define what it means for M to be a torsion free module.

(b) Define what it means for M to be a torsion free module. (c) Suppose that M is finitely generated over R and let M_{t} be the torsion submodule of M. Show that M = M_{t} ⊕ N where N is a free module. (You may use whatever facts you know about free modules over a PID, but be sure to clearly state them when you use them.)

5. Let F be a field and f(x) ∈ F [x] a polynomial with coeffecients in F. Show that there exists a finite extension K of F in which f(x) has a root. (Construct the extension.)

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6. Let f(x) = x^{4} − 25 ∈ ℚ[x].

(a) Determine the splitting field E of f(x) and all intermediate fields ℚ ⊂ F ⊂ E.

(b) Determine the Galois G group of f(x) and all of its subgroups H.

(c) Explicitly exhibit the Galois correspondence between all intermediate fields ℚ ⊂ F ⊂ E and all subgroups H ⊂ G.