2021 Fall Algebra

Problem 1.

Let be a vector space over a field . Let be a linear transformation. Let denote eigenvectors of with distinct eigenvalues. Prove that the set is linearly independent.

Problem 2.

(a) Let denote an odd prime. Prove that, up to isomorphism, there are precisely two groups of order . (The same holds if , but you don't need to show that.)

(b) Let be a positive integer and be a prime. What is the order of a Sylow -subgroup of ? Give a brief explanation.

Problem 3.

Assume is a group and , are normal subgroups of such that the groups and are abelian. You may use without explanation that is a normal subgroup of . Prove that is also abelian.

Problem 4.

(a) Let denote a prime and let denote a subgroup which acts transitively on . Show that contains a -cycle.

(b) Give an example of a subgroup of which acts transitively on and which does not contain a 4-cycle.

Problem 5.

Let be any element. Prove that the principal ideal is not a maximal ideal in .

Problem 6.

Let be a field with elements. Prove that 3 is not a square of an element in .

Problem 7.

Let be a prime and .

(a) For which does there exist a subfield of such that is a Galois extension with . Justify your answer.

(b) Let be an extension as in part (a). Prove that .

Problem 8.

Let be an integral domain and let be an -module. Recall that denotes all such that there exists such that . An -module is called torsion-free if .

(a) Prove that is an -submodule of .

(b) Prove that is torsion-free.

Problem 9.

For each of the following, either give an example or briefly explain why no such example exists.

(a) A group and a subgroup such that is simple and is not simple.

(b) A non-abelian group such that the map given by is a group homomorphism.

(c) A real matrix such that , where is the identity matrix.

(d) A field and a polynomial such that the splitting field of over is not a Galois extension of .