Algebra Prelim, Spring 1989

Do all problems

1. Show that a group of order 540 cannot be simple.
2. Compute the Galois group of 5x5 + 3x4 +15 over

3. Let R and S be domains and let be an epimorphism. Which of the following statements are true? (Prove or give a counterexample.)
1. If R is a PID, then S is a PID.
2. If R is a UFD, then S is a UFD.
3. If and R is a PID, then S is a field.
4. Let K be a field and let be a polynomial.
1. Let be distinct zeros of f in K. Prove that there exists such that

2. 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 have

Deduce that p divides (p-1)! + 1.

5. Let G be a group of finite order and let F be the intersection of all maximal subgroups of G.
1. Prove that .
2. If and FH = G, prove that H=G.
3. If S is a Sylow subgroup of F and , prove that xSx-1 = fSf-1 for some . Deduce that .
6. Which of the following statements are true? (Prove or give counterexample.)
1. If K/F and E/K are finite Galois extensions, then E/F is a finite Galois extension.
2. Let be irreducible, let be a root of f, and let be a root of g. If , then .
7. 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.

1. Show that for some and some R-submodule N of Rn.
2. Deduce that if L is a submodule of M, then where K is a submodule of Rn containing N.
3. Conclude that all finitely generated R-modules are Noetherian.
8. Determine the matrices in commuting with .