Algebra Prelim, August 2010
Do all problems
- Let G be a group of order 105 with a normal Sylow 3-subgroup. Show
that G is abelian.
- Find the Galois group of
f (x) = x4 -2x2 - 2 over
describing its generators explicitly as permutations of the roots of
- Let p be a prime. Prove that the extension
Fpn⊃Fp has Galois group generated by the Frobenius
σ : Fpn -> Fpn
σ(a) = ap for all
- Show that there is no 3 by 3 matrix A with entries in
such that A8 = I but
- Let R be an integral domain. A nonzero nonunit element
prime if p | ab implies p | a or p | b. A
nonzero nonunit element
p∈R is irreducible if p = ab
implies a or b is a unit. Show that
- Every prime is irreducible.
- If R is a UFD, then every irreducible is prime.
- Let R be the ring
Z/6Z and let I be the ideal
3Z/6Z. Prove that
- Let S be a multiplicatively closed nonempty subset of the
commutative ring R with a 1. Assume that
- Show that if R is a PID, then S-1R is a PID.
- Show that if R is a UFD, then S-1R is a UFD.
- Let R be a commutative ring with a 1.
- Show that if
x∈R is nilpotent and
y∈R is a unit in R,
then x + y is a unit in R.
f = a0 + a1x + a2x2 + ... + anxn∈R[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.