Algebra Prelim, August 1997

1. Let R be a commutative ring with unity and let I, J be ideals of R.
1. Prove that the product IJ = {x R | x = Si = 1naibi with ai I and bi J} is an ideal of R.

2. Prove that IJ I J.

3. If I + J = R, prove that IJ = I J.

4. If IJ = I J for all ideals of R and R is an integral domain, prove that R is a field. (Hint: let I = Ra where a 0 be a principal ideal or R.)

2. Let F be a finite Galois extension of the field K with Gal(F/K) S5.
1. Show that there are more than 40 fields strictly between F and K.

2. Show that there is a unique proper subfield E of F with EK such that E/K is a Galois extension. Determine [E : K] and describe Gal(E/K) up to isomorphism.

3. Let G be a group of order 455.
1. Prove that G is not simple.

2. Prove that G is cyclic.

4. Let R be a PID, let n be a positive integer, and let A and B be finitely generated R-modules. If An Bn, prove that A B. (An denotes the direct sum of n copies of A.)

5. Let P be a finitely generated projective -module. If P is also injective, prove that P = 0.

6. Let A, B be abelian groups, and let m be a positive integer. Prove that A (B/mB) (A B)/m(A B).

7. Prove that a group of order 588 is solvable.

8. Let K = ( + ,).
1. Determine [K : ].

2. Compute Gal(K/).

Peter Linnell
1999-06-16