Algebra Preliminary Exam, Spring 1986
- Let F be the free group on the set ,
and let H be the normal subgroup of F generated by
. Prove that F/H is
isomorphic to the free abelian group on X.
- Let G be an abelian group of order p6, and let . Suppose that |H|=p2. Give all possible such
- Prove that S4 is solvable.
- In S5, how many Sylow subgroups of each type are there?
- Let R be a PID and let S be a multiplicatively closed
subset. Show that S-1R is a PID.
- Give an example of a PID with exactly 3 nonassociate irreducible
- Let be the set of maximal ideals in
a commutative ring R with identity. Set . For , prove that if and only if 1+rs
is a unit for all .
- Let be a root of x3+4x+2.
- Find a basis for over . Justify
y our answer.
- Express in terms of the basis.
- Express in
terms of the basis.
- Let K be a subfield of the field L, and let
such that is odd. Prove that .
Wed Jul 31 14:18:12 EDT 1996