Masters Comprehensive Exam Modern Algebra January 1993

1. (a) If G is a group of order p^{n}, with p a prime, show that G has a non-trivial center.

(b) If G is a group of order p^{2}, with p a prime, show that G is abelian.

(c) Are all groups G of order p^{3}, with p a prime, abelian? (Prove or give a counter-example.)

2. (a) List all abelian groups of order 28 (up to isomorphism).

(b) Can a group of order 28 be simple? Prove your answer.

(c) Construct 2 non-abelian groups of order 28. How many non-abelian groups of order 28 do you think there are?

3. Let R be a commutative ring with unit. Recall that an ideal P ⊂ R is called prime if ab ∈ P implies a ∈ P or b ∈ P.

(a) Show that any maximal ideal of R must be prime.

(b) If R is a PID, show that all prime ideals are maximal.

(c) Give an example of a commutative ring R and an ideal I ⊂ R such that I is prime but not maximal.

4. Let M be a free module of finite rank over a PID R.

(a) Show that every submodule N ⊂ M is also free.

(b) Is it true that every quotient module of the free module M is also free? If so, prove. If not, give a counter example.

5. (a) If V is a finite dimensional vector space over a field F and V^{∗} is its dual vector space, show that V ≃ V^{∗}.

(b) Give an example of a vector space over ℚ which is not finite dimensional.

6. (a) Compute the Galois group G of f(x) = x^{3} − 5 over ℚ.

(b) If F ⊃ ℚ is the splitting field of this polynomial, write down all intermediate fields F ⊃ E ⊃ ℚ.

(over)

Typeset by AMS-TEX

7. Let F ⊃ E be an extension of fields.

(a) Define what it means for F to be an algebraic extension of E.

(b) If (F : E) < ∞, show that F is an algebraic extension of E.

(c) Do infinite algebraic extensions exist? If not, prove it. If so, give an example.