2022 Fall Algebra

Problem 1.

Let be a group, be the set of all homomorphisms and be the set of all automorphisms of . Consider the group actions of on , defined by where is the inner automorphism .

(a) For the action on : Let . Determine the relationship between the stabilizer of under this action and teh centralizer .

(b) For the action on : Prove that this action is transitive if and only if all automorphisms of are inner.

Problem 2.

Let be a finite abelian group. Show that is cyclic if and only if for any prime it has either 0 or elements of order .

Problem 3.

Let denote the field with elements. Find the order of a Sylow -subgroup of the group and give an example of a Sylow -subgroup.

Problem 4.

Let be a commutative ring with identity and an ideal such that . Show that there are infinitely many maximal ideals in that contain .

Problem 5.

Let be a unique factorization domain in which there exists a unique irreducible element up to associates.

(a) Prove that is a principal ideal domain.

(b) Show that has a unique maximal ideal and a unique non-maximal prime ideal.

Problem 6.

Let be a submodule in a module over a commutative ring which is a PID. Assuming that is free of finite rank, show that is also free of finite rank.

Problem 7.

Consdier the ring and the ideal of generated by elements 2 and . Notive that the rings and are isomorphic.

(a) Prove that the map defined by

is -balanced (i.e. it is additive in each component, and for all and ).

(b) Prove that the element in .

(c) Prove that the element is annihilated by both 2 adn in .

(d) Prove that the submodule of generated by the element is isomorphic to .

Problem 8.

Assume is a field and is a finite field extension of of odd degree. Prove that .

Problem 9.

Let be a field with 64 elements and . Suppose that has order in the multiplicative group . Show that generates the field over , i.e. .

Problem 10.

Let be a Galois extension with Galois group isomorphic to and the fixed point subfield of the cyclic subgroup generated by the 5-cycle . Find the number of different subfields in which are isomorphic to (including the case ).