Assignment 6

Due Thursday, April 20, 2023

This is a bonus assignment. It is possible to get 100% in the course from only the first five assignments.

This assignment is “out of” 100 points, but there are far more than 100 points available. At the instructor’s discretion some “overflow” above 100 may be counted towards your final grade at the end of the course, but you should not expect this.

You are not expected to write up a full solution to every problem, but you are expected to at least think about every problem. Writing up every single problem on the assignment is probably not a good use of your time.

Throughout, the notation $Q_{a, b}$ denotes the quaternion $k$ -algebra $Q_{a, b} = k ⟨ i, j ∣ i^{2} = a, j^{2} = b, i j = - i j ⟩$ where $k$ is a field and $a, b$ are non-zero elements of $k$ .

Errata (2023/04/18):

Fixed typo in 3.

Problem 1 (20 points)

Prove that if $R$ is a ring such that every left $R$ -module is free, then $R$ is a division algebra.

Problem 2 (20 points)

Classify the two-sided ideals of $M_{2} (Z)$ .

Problem 3 (40 points)

Determine the primes $p$ such that $Q_{- 1, p}$ splits over $k = Q$ .

Problem 4

Let $k$ be a field and $a, b, c, d$ non-zero elements of $k$ .

(a) (20 points)

Find an explicit bijection $Q_{1, 1} ≅ M_{2} (k)$ .

(b) (10 points)

Prove that $Q_{a, b} ≅ Q_{b, a}$ .

Prove that $Q_{a, b} ≅ Q_{a c^{2}, b d^{2}}$ .

(d) (30 points)

Prove that $Q_{1, b} ≅ Q_{1, 1}$ .

(e) (30 points)

Prove that $Q_{a, 1 - a} ≅ Q_{1, 1}$ .

Problem 5

In each of the following cases, determine the Wedderburn decomposition of the group ring $k G$ (in other words, write $k G$ as a product of simple $k$ -algebras.

(a) (20 points)

The group ring $Q C_{3}$ where $C_{3}$ is the cyclic group of order $3$ .

(b) (20 points)

The group ring $Q S_{3}$ where $S_{3}$ is symmetric group on $3$ letters.

The group ring $Q D_{8}$ where $D_{8}$ is the dihedral group on $4$ letters.

(d) (40 points)

The group ring $Q S_{4}$ where $S_{4}$ is symmetric group on $4$ letters.

(e) (30 points)

The group ring $Q Q_{8}$ where $Q_{8}$ is quaternionic group of order $8$ .

(f) (80 points)

The group ring $Q G$ where $G = (Z / 3 Z) ⋊_{ϕ} (Z / 4 Z)$ where $ϕ : Z / 4 Z \to Aut (Z / 3 Z)$ is the unique non-trivial homomorphism.

MATH 742 - 001 - Spring 2023

MATH 742 - 001 - Spring 2023

Assignment 6