2015 Spring Algebra

Problem 1.

Prove that every finite group of order greater than 2 has a nontrivial automorphism.

Problem 2.

In this problem there is no need to justify your answers.

(a) Define UFD.

(b) Define PID.

(c) For the properties "UFD" and "PID", give an example of a commutative integral domain that

i. satisfies both properties

ii. satisfies one property but not the other

iii. satisfies neither property

Problem 3.

(a) Prove that is not Galois over , where is an indeterminate.

(b) Find the Galois closure of over and determine the Galois group both as an abstract group and as a set of explicit automorphisms. (Fully justify.)

Problem 4.

Let be a commutative ring with multiplicative identity. An element is called nilpotent if there exists a postive integer such that .

(a) Prove that every nilpotent element lies in every prime ideal.

(b) Assume that every element of is either nilpotent or a unit. Prove that has a unique prime ideal.

Problem 5.

For every positive integer , denote by a cyclic group of order and by a dihedral group of order , so that

where has order , has order 2 and .

(a) In the notation explained above, prove that every subgroup of is normal in .

(b) If with odd, prove that .

(c) Is ? Justify your answer.

Problem 6.

Suppose that and are prime numbers with . Prove that no group of order is simple.

Problem 7.

Determine the maximal ideals of the following rings (fully justify):

(a)

(b)

Problem 8.

Find two matrices having the same characteristic polynomials and minimal polynomials but different Jordan canonical forms. Fully justify.

Problem 9.

(a) What does it mean for a field to be perfect?

(b) Give an example of a perfect field. (No need to justify your answer.)

(c) Give an example of a field that is not perfect. (No need to justify your answer.)

Problem 10.

(a) Classify the conjugacy classes of the symmetric group and justify.

(b) Construct the character table of .