2019 Fall Algebra

Problem 1.

Do there exist groups , such that is a semi-direct product of and ?

Problem 2.

Let be a finite group of order where are primes. Prove that either has a normal Sylow -subgroup of is isomorphic to , the alternating group on 4 letters.

Problem 3.

Let be a ring with . Let be the set of all nilpotent elements of . Prove the following:

(a) If is commutative then is an ideal of and is equal to the intersection of all prime ideals of .

(b) Give an example of a non-commutative ring such that is not an ideal of .

Problem 4.

Work with the ring .

(a) Is a prime in ?

(b) Identify the quotient . Which ring is it?

You may want to use the fact that has a Euclidean norm.

Problem 5.

Let be a commutative ring with . Consider two maximal ideals of .

(a) Prove that every elemnt of is either a unit or a zero divisor.

(b) Prove that always has a unit and at least two zero divisors.

(c) Prove that has infinitely many units if and only if it has infinitely many zero divisors.

Recall that a maximal ideal of is, by definition, a proper ideal of .

Problem 6.

Assume are field extensions such that both and are algebraic. Prove that is algebraic.

Problem 7.

Calculate the Galois group of over .

Problem 8.

Let be a field extension of such that is cyclic of order 4. Prove that .

Problem 9.

Let be a finite dimensional vector space over and let be a linear transformation such that but .

(a) What are the possible degrees of the minimal polynomial of ?

(b) Assume there are non-similar linear transformations such that but . What is the smallest possible dimension of ?

Problem 10.

Prove that if is a square matrix over then and its transpose are conjugate over .