2017 Fall Algebra

Problem 1.

Suppose has eigenvalues -1 and 2 (and no other eigenvalues). Let denote the characteristic polynomial of , and the minimal polynomial. Which pairs can occur? For each pair that can orccur, find an explicit example of a matrix with those characteristic and minimal polynomials.

Problem 2.

Prove that no group of order 150 is simple.

Problem 3.

Suppose is an abelian group, and , are subgroups. Either prove the following statement, or find a counterexample:

Problem 4.

Determine up to isomorphism all -modules of order 4.

Problem 5.

Suppose that is a field of characteristic 0, and is the splitting field of the irreducible polynomial . Prove that if is abelian, and if is a root of , then .

Problem 6.

(a) Let . What is ?

(b) Let . What is ?

Problem 7.

Which of the following ideals of are prime? Which are maximal? Justify your answer.

Problem 8.

If is a finite nromal subgroup of a group , , and has an element of order , then so does .

Problem 9.

Indicate whether each of the following statements is True or False, and give a brief justification.

(a) Every commutative ring with identity, with exactly 200 elements, has zero divisors.

(b) For every prime there is a nonzero homomorphism from .

(c) The center of a non-abelian group is always properly contained in some abelian subgroup.

(d) If is a subfield of and is isomorphic to as fields, then .

(e) For every integral domain and every -module , the set of torsion elements is a sub-module. (We say is a torsion element is there is a nonzero such that .)