Ph.D. Qualifying Exam–Algebra
Fall 2004
Answer as many questions as you can. Give an appropriate amount of detail for each proof.
Your judgment in this matter is an important part of the exam.
1. Prove that if Ris a commutative ring with 1, then every maximal ideal of Ris prime.
2. Let pand qbe distinct primes. Prove that any group of order pq is solvable.
3. Define the commutator subgroup G0of a group G, and prove that if Nis a normal
subgroup of Gsuch that G/N is abelian, then G0is a subgroup of N.
4. Determine (with proof) a complete set of representatives of the conjugacy classes of
the group GL3(F2). Be sure that your list has no repetition.
5. Let mand nbe positive integers. Compute (with justification) (Z/mZ)⊗Z(Z/nZ).
6. Let Rbe a commutative ring with 1. Prove carefully that every proper ideal of Ris
contained in a maximal ideal of R.
7. Let Rbe a commutative ring with 1, and let I1, . . . , Inbe ideals of R. If
J=I1∩I2∩ · ·· ∩ In
is a prime ideal of R, show that at least one of the ideals Ik, with k∈ {1, . . . , n}, is
prime.
8. Find the Galois group of x5−4x3+ 2 over Q.
9. Prove that no finite field is algebraically closed.
10. Determine (with proof) the number of irreducible degree 6 polynomials in F2[x].
1