**Algebra Prelim, Spring 1989**

Do all problems

- Show that a group of order 540 cannot be simple.
- Compute the Galois group of 5
*x*^{5}+ 3*x*^{4}+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.

- If
- 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 haveDeduce that

*p*divides (*p*-1)! + 1.

- Let be distinct zeros of
- 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
counterexample.)
- 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 .

- If
- 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*R*^{n}are Noetherian for all (where*R*^{n}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*R*^{n}. - Deduce that if
*L*is a submodule of*M*, then where*K*is a submodule of*R*^{n}containing*N*. - Conclude that all finitely generated
*R*-modules are Noetherian.

- Show that for some and some
- Determine the matrices in commuting with
.

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Wed Jul 31 09:21:39 EDT 1996