Algebra Qualifying Examination
January, 2005
Directions:
1. Answer all questions. (Total possible is 100 points.)
2. Start each question on a new sheet of paper.
3. Write only on one side of each sheet of paper.
Policy on Misprints:
The Qualifying Exam Committee tries to proofread the exams as carefully as possible. Nevertheless, the exam may contain a few misprints. If you are convinced a problem has been stated incorrectly, indicate your interpretation in writing your answer. In such cases, do not interpret the problem in such a way that it becomes trivial.
Notes: All rings are unitary. All modules are unitary.
Q is the rationals, R the reals, C the complexes, and Z the integers.
1. (10 points) Let T : R
3
→ R
3 be a linear transformation. Prove that T has a one dimensional invariant subspace and a two dimensional invariant subspace.
2. (10 points) Show that there is no simple group of order 160.
3. (20 points) Let G be a finite group having the property that, for all x, y ∈ G , there exists an integer j (depending on x and y ) such that xyx
−
1 = y j .
a. Prove that G is the direct product of its Sylow subgroups.
b. Prove that, if x, y ∈ G have prime order p , then xy = yx .
4. Let R be a ring with identity 1 and consider the following two conditions:
(I) If a, b ∈ R and ab = 0, then ba = 0.
(II) If a, b ∈ R and ab = 1, then ba = 1.
a. (10 points) Show that (I) implies (II).
b. (5 points) Show by example that (II) does not imply (I).

5. (5 points) Give an example of a ring R and a projective R − module P which is not free. Justify your answer.
6. a. (10 points) Let M be a left module over a commutative ring R , and let I be an ideal of R . Prove that ( R/I ) ⊗
R
M is isomorphic to M/IM as a left R -module.
b. (5 points) Consider any Abelian groups in this part as Z -modules. Calculate as explicitly as possible the Z -module ( Z / 2 Z ) ⊗ ( Z / 8 Z ). Your solution should demonstrate the validity of your final answer.
7. Suppose that F is a field of characteristic p and that f ( x ) = x p
− x − c for some c ∈ F . Suppose that α is a root of f ( x ) in some extension field of E of F .
a. (10 points) Prove that α + 1 is also a root of f ( x ).
b. (5 points) Prove that F ( α ) is a splitting field of f ( x ) over F .
c. (10 points) Determine all possible Galois groups G ( F ( α ) /F ) and show when each occurs.