Algebra Preliminary Exam, Fall 1981
Instructions: do all eight problems
Notation: = integers, = rational numbers
- Let G be a finite group.
- Let A and B be subgroups of G such that and
AB = G. Prove that .
- Let H be a subgroup of G such that [G:H] = 2 and let
. If m is the number of conjugates of h in G and n is
the number of conjugates of h in H, prove that either n=m or
- If the order of G is 105 and H is a subgroup of G of
order 35, prove that .
- Let P be a nonnormal p-Sylow subgroup of the finite group
G. If is the normalizer of P in G, prove that
is nonnormal in G.
- Let F1 and F2 be finite fields of orders q1 and q2.
- Prove that qi is a power of a prime, say for i=1,2.
- If , prove that p1=p2 and that
is a divisor of .
- Prove that x4-2 is irreducible over .
- Let K be the splitting field of x4-2. Prove that the
Galois group of K over , , is of order
- Exhibit the correspondence (given by the Fundamental Theorem of
Galois theory) between the subgroups of and the
intermediate fields between and K.
- State Nakayama's lemma.
- Let R be a local commutative ring with maximal ideal M.
Let X be a finitely generated R-module. Show that if X/MX can
be generated by n elements, then so can X.
- Construct an example of finitely generated
nonzero abelian groups A
and B so that .
- If A and B are finitely generated abelian groups
such that and , prove that and .
- Let R be a commutative ring and let A be an ideal of R.
Define the radical of A, denoted , by for some positive integer n}. You may assume
that is an ideal of R and that .
- If P is a prime ideal of R such that , prove
that and as a consequence show that .
- Prove that is the set of nilpotent elements of
R/A. (An element r is nilpotent if rn = 0 for some positive
Thu Aug 1 09:27:17 EDT 1996