Algebra Prelim, August 2009
Do all problems
- Let p be a prime, let n be a non-negative integer, and let S be
a set of order pn. Suppose G is a finite group that acts
transitively by permutations on S (so if
s, t∈S, then there
g∈G such that gs = t) and let P be a Sylow
p-subgroup of G. Prove that P acts transitively on S.
- Prove that there is no simple group of order
448 = 7*64.
- Prove that x2 + 1 is irreducible in
- Prove that
x3 +3x2 - 9x + 12 is irreducible in
- Let R be a PID, let M be a finitely generated right R-module,
and let N be an R-submodule of M. Prove that there exists an
R-submodule L of M and
0≠r∈R such that
L∩N = 0
Mr⊆L + N.
- Let R be a ring and suppose we are given a commutative diagram
of R-modules and homomorphisms,
A -------------> B
\/ j \/
C -------------> D
where f is onto and j is one-to-one. Prove that there exists
a unique R-module homomorphism
k : B -> C such that the
resulting diagram commutes (so kf = g and jk = h).
- Compute the isomorphism type of the Galois group of
x4 -2x2 + 9
- Let R be a commutative ring with a 1 and let I, J be ideals of
R. Prove that
R/I⊗RR/J≌R/(I + J) as R-modules.