Groups and Rings 2014/2015

Syllabus

News

The location of the exam depends on the first letter of your last name: A‑F room D32, G‑R room D33, S‑Ö room D34.

The exam will take place March 16 (8:00-13:00). Anyone who plans to take the exam has to register. Registration is open until Feb 22, (24:00).

Homework 12 due Mar 6, via e-mail or in the homework box at the expedition of the mathematics department

Syllabus for Part I: Groups

Nov 4: Lecture 1. Chapter 1.1–1.2(p.25)

Introduction to groups.

Nov 7: Exercises

Suggested exercises: Chapter 1.1: 1,5,7,9,11,20,22,28,34
Chapter 1.2: 10

Nov 11: Lecture 2. Chapter 1.2–1.4

Dihedral groups, symmetric groups, matrix groups.

Nov 14: Exercises

Suggested exercises: Chapter 1.2: 3,7,15
Chapter 1.3: 1,3,6,9,11,15
Chapter 1.4: 2,7

Nov 20: Lecture 3. Chapter 1.6,2.1–2.2, 2.4–2.5

Homomorphisms and isomorphisms, subgroups, centralizers, normalizers, generation.

Nov 21: Exercises

Suggested exercises: Chapter 1.6: 2,4,9,17
Chapter 2.1: 1,3,6,8
Chapter 2.2: 10,14
Chapter 2.4: 3,8,13

Nov 25: Lecture 4. Chapter 2.3, 3.1

Cyclic groups. Cosets, normal subgroups and quotient groups.

Nov 28: Exercises

Suggested exercises: Chapter 2.3: 23
Chapter 2.3: 1,3
Chapter 3.1: 1,3,8,10,21

Dec 3: Lecture 5. Chapter 3.2, 1.7, 2.2 (p.51–52)

Group actions and Lagrange's theorem

Dec 4: Exercises

Suggested exercises: Chapter 3.1: 24,29,33,36
Chapter 2.2: 7,13
Chapter 3.2: 1,2,16

Dec 10: Lecture 6. Chapter 3.3, 4.2, 4.3

Isomorphism theorems, orbits and stabilizers, the class equation

Dec 11: Exercises

Suggested Exercises: Chapter 3.3: 1,3
Chapter 4.2: 2,8
Chapter 4.3: 2,3,5,6,10,13,25,30

Dec 18: Lecture 7. Chapter 4.4–4.5

The Sylow theorems and applications.

Dec 19: Exercises

Syllabus for Part II: Rings

Jan 20: Lecture 8. Chapter 7.1-7.2, 9.1.

Rings, integral domains, fields. Examples

Jan 23: Exercises

Suggested Exercises: Chapter 7.1: 5, 6, 7, 8, 11, 15, 16
Chapter 7.2: 1, 3, 5
Chapter 9.1:

Jan 27: Lecture 9. Chapter 7.3–7.4.

Homomorphisms, ideals, quotient rings, maximal ideals and prime ideals.

Jan 30: Exercises

Suggested Exercises: Chapter 7.3: 4, 6, 8, 13, 17, 19, 29
Chapter 7.4: 5, 7, 8, 10, 15, 17

Fab 3: Lecture 10. Chapter 7.5–7.6.

Rings of fractions. The Chinese Remainder Theorem.

Fab 6: Exercises

Suggested Exercises: Chapter 7.5: 2, 3
Chapter 7.6: 5, 6, 7

Fab 10: Lecture 11. Chapter 7.5 and 8.1

Rings of fractions. Euclidean domains.

Fab 12: Exercises

Suggested Exercises: Chapter 7.5: 2, 3
Chapter 8.1: 1, 6, 7, 8, 10

Fab 17: Lecture 12. Chapter 8.2-8.3, 9.2-9.3

Principal ideal domains, unique factorization domains, and more on polynomial rings.

Fab 20: Exercises

Suggested Exercises: Chapter 8.2: 5
Chapter 8.3: 5,10
Chapter 9.2: 3, 9
Chapter 9.3: 1, 3

Fab 24: Lecture 13. Chapter 9.2-9.4

polynomial rings and irreducibility.

Fab 27: Exercises

Suggested Exercises: Chapter 9.3: 1, 3
Chapter 9.4: 1, 2, 7, 8, 9