Algebra Prelim, August 2010

Do all problems

1. Let G be a group of order 105 with a normal Sylow 3-subgroup. Show that G is abelian.

2. Find the Galois group of f (x) = x4 -2x2 - 2 over Q, describing its generators explicitly as permutations of the roots of f.

3. Let p be a prime. Prove that the extension FpnFp has Galois group generated by the Frobenius automorphism σ : Fpn -> Fpn given by σ(a) = ap for all aFpn.

4. Show that there is no 3 by 3 matrix A with entries in Q, such that A8 = I but A4I.

5. Let R be an integral domain. A nonzero nonunit element pR is prime if p | ab implies p | a or p | b. A nonzero nonunit element pR is irreducible if p = ab implies a or b is a unit. Show that
(a)
Every prime is irreducible.

(b)
If R is a UFD, then every irreducible is prime.

6. Let R be the ring Z/6Z and let I be the ideal 3Z/6Z. Prove that IRII as R-modules.

7. Let S be a multiplicatively closed nonempty subset of the commutative ring R with a 1. Assume that 0 S.
(a)
Show that if R is a PID, then S-1R is a PID.

(b)
Show that if R is a UFD, then S-1R is a UFD.

8. Let R be a commutative ring with a 1.
(a)
Show that if xR is nilpotent and yR is a unit in R, then x + y is a unit in R.

(b)
Let f = a0 + a1x + a2x2 + ... + anxnR[x]. Show that f is a unit in R[x] if an only if a0 is a unit in R and ai is nilpotent for i > 0.

Peter Linnell 2010-12-20