2015 Fall Algebra

Problem 1.

(a) Define prime ideal.

(b) Define maximal ideal.

(c) Give an example of a ring and ideals , , and of such that for the properties "prime ideal" and "maximal ideal" of ,

i. satisfies both properties,

ii. satisfies neither property,

iii. satisfies one property but not the other.

Justify your answers.

Problem 2.

Show that if a group has only finitely many subgroups then is a finite group.

Problem 3.

Let be an matrix with entries in such that .

(a) Prove that is even.

(b) Prove that is diagonalizable over and describe the corresponding diagonal matrices.

Problem 4.

Let be a group of order 70. Prove that has a normal subgroup of order 35.

Problem 5.

Construct a Galois extension of satisfying , the dihedral group of order 8. Fully justify.

Problem 6.

Let be a field. Prove that every ideal of is principal.

Problem 7.

Give an example of a module over a ring such that is not finitely generated as an -module. Prove that it is not finitely generated as an -module.

Problem 8.

Suppose is a normal subgroup of a finite group .

(a) Prove or disprove: If has order 2, then is a subgroup of the center of .

(b) Prove or disprove: If has order 3, then is a subgroup of the center of .

Problem 9.

(a) What does it mean for a representation to be irreducible?

(b) Suppose is a prime. Let and let be a representation. Show that is reducible.

Problem 10.

(a) Compute the order of . (Justify your reasoning.)

(b) Compute the order of . (Justify your reasoning)

(c) Show that is not a UFD.