Algebra Prelim, Fall 1992
- Prove that any group of order 765 is abelian.
- Let G be a finite group and let N be a normal subgroup of
G. Prove that G is solvable if and only if both N and G/N are
- Prove that if G is a subgroup of Sn, then either
- Prove that if and G is a normal subgroup of Sn,
then , An or Sn.
- Prove that if , then Sn has no subgroup of index
- Let f(x) be an irreducible polynomial over with
splitting field K. If the Galois group of is abelian,
prove that for any roots of f(x) in K, we have
- Let K be a field and be a separable
irreducible polynomial of degree 4, and let E be a splitting field
for f(x) over K. If is a root of f(x) and
, prove that there exists a subfield F of L with [F:K]
= 2 if and only if the Galois group of E/K is not isomorphic to
either A4 or S4.
- If G, H and K are finitely generated abelian groups with
, prove that .
- Give an example to show that part (a) is false if G is not
- Let R be an integral domain. A nonzero element of R
is a prime if implies that
either or .
A nonzero element is irreducible if implies
that either a or b is a unit.
- Prove that every prime is irreducible.
- If R is a UFD, prove that every irreducible is prime.
- Let R be a commutative ring with a 1, and let M be a cyclic
- Prove that M is isomophic to R/I for some ideal I of R.
- If N is any R-module, prove that is
isomorphic to N/IN for some ideal I of R.
Tue Jul 30 17:01:19 EDT 1996