2022 Winter Algebra

Problem 1.

Assume is a normal subgroup of a group . Let denote the set of left cosets of in . Prove that the following operation is well-defined:

Problem 2.

Let be an integer. You may use without proof that is the only non-trivial proper normal subgroup of . Let be a subgroup of such that . Prove that .

Problem 3.

Prove that if is a group of order 12, then is isomorphic to a semidirect product where are proper non-trivial subgroups of .

Problem 4.

Assume is an matrix over some field , and assume is a vector such that and for some , where denotes the identity matrix. Prove that is not diagonalizable.

Problem 5.

Let be a ring and assume is a prime ideal in . Assume satisfies the descending chain condition, meaning that for every countably infinite descending chain in , there exists such that for all integers . Prove that is a maximal ideal in .

Problem 6.

Consider the subring of defined as follows: . Prove that the element cannot be expressed as a product of irreducible elements.

Problem 7.

Determine the structure (as a direct product of cyclic groups) of the group of units of the ring .

Problem 8.

Let be an integer, and let denote a primitive -th root of unity.

(a) Give an explicit bijection between and . You don't have to prove it's a bijection.

(b) What is the degree of ? Justify your answer.

Problem 9.

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

(a) Finite order elements in a group such that has infinite order.

(b) A surjective group homomorphism .

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

(d) A tower of fields such that and are Galois extensions but is not a Galois extension.