Algebra Prelim, Spring 1989
- Show that a group of order 540 cannot be simple.
- Compute the Galois group of 5x5 + 3x4 +15 over
- Let R and S be domains and let be an
epimorphism. Which of the following statements are true? (Prove or
give a counterexample.)
- If R is a PID, then S is a PID.
- If R is a UFD, then S is a UFD.
- If and R is a PID, then S is a field.
- Let K be a field and let be a polynomial.
- Let be distinct zeros of f in K.
Prove that there exists such that
- Let p be a prime number and let be the finite
field with p elements. For each integer m, let denote
its residue class in K. Prove that as polynomials in K[x], we
Deduce that p divides (p-1)! + 1.
- Let G be a group of finite order and let F be the
intersection of all maximal subgroups of G.
- Prove that .
- If and FH = G, prove that H=G.
- If S is a Sylow subgroup of F and , prove that
xSx-1 = fSf-1 for some . Deduce that .
- Which of the following statements are true? (Prove or give
- If K/F and E/K are finite Galois extensions, then E/F is
a finite Galois extension.
- Let be irreducible, let be a
root of f, and let be a root of g. If , then .
- Let R be a commutative ring. Define what is meant by saying
that an R-module is Noetherian.
Suppose R has the property that the R-modules Rn are Noetherian
for all (where Rn denotes the direct sum of n
copies of R, and . Let M be a
finitely generated R-module.
- Show that for some and some
R-submodule N of Rn.
- Deduce that if L is a submodule of M, then
where K is a submodule of Rn containing N.
- Conclude that all finitely generated R-modules are
- Determine the matrices in commuting with
Wed Jul 31 09:21:39 EDT 1996