Fall 2007 : MATH566 Abstract Algebra I

  1. General information
  2. Textbook
  3. Homework
  4. Exams
  5. Grading scheme
  6. Course syllabus

General information

Textbook

The textbook for this course is :

dummit.jpeg

David S. Dummit and Richard M. Foote "Abstract Algebra" 3rd edition, Wiley 2003.

It is available from the campus bookstore. The authors' list of errata are available online.

Homework

There will be weekly homework, set from the textbook. The homework will be announced on Friday on this webpage (next to the syllabus), and must be handed in the following Wednesday at the beginning of the lecture.

Exams

There will be one midterm and one final exam. They will both be held in the same room as the lectures (ENGRG E203).

If you are unable to attend any of these exams because of a legitimate reason (for example, it clashes with an exam for another course), then you must let the instructor know at least one week in advance.

Grading scheme

Course grades will be determined based on homework (50%), the midterm exam (20%), and the final exam (30%). In addition, this is a "portfolio course" for graduate students in the mathematics department: your result in the final exam alone will count towards your PhD qualification. Topics which may appear on the final exam should be drawn from this list.

Course syllabus

The syllabus below will be updated as the semester progresses.

Week

Material covered (with page numbers)

Homework

Aug 20-24

Chapter 1 : Introduction to Groups (13-45)

  • definitions
  • examples (dihedral, symmetric, matrix, and quaternion groups)
  • homomorphisms and isomorphisms
  • group actions

Please write out solutions to half of these (your choice):
Page 21, exercises 7, 9, 25, 31.
Page 27, exercises 3, 7, 12, 17.
Page 32, exercises 1, 12, 14.
Page 35, exercise 7.
Page 36, exercise 3.
Page 39, exercises 6, 9, 14, 17.
Page 44, exercises 4, 8, 16, 21, 23.
Due Friday 31st August, or thereabouts.

Aug 27-31

No classes this week

.

Sept 3-7

Chapter 2 : Subgroups (46-72)

  • definitions and examples
  • centralizers and normalizers
  • cyclic groups

Page 48, exercises 1d, 2c, 4, 6.
Page 52, exercises 5, 6, 10.
Page 60, exercises 13, 16, 20, 23.
Due Wednesday 12th September.

Sept 10-14

Chapter 2 : Subgroups (46-72)

  • subgroups generated by subsets
  • the lattice of subgroups
Chapter 3 : Quotients and Homomorphisms (73-111)
  • definitions and examples

Page 65, exercises 3, 10, 12, 15.
Page 71, exercises 7, 11.
Page 85, exercises 17, 25, 31, 34, 39.
Page 95, exercise 4.
Due Wednesday 19th September.

Sept 17-21

Chapter 3 : Quotients and Homomorphisms (73-111)

  • cosets and Lagrange's Theorem
  • the Isomorphism Theorems

Page 95, exercises 12, 14, 19.
Page 101, exercises 3, 5, 8, 9.
Due Wednesday 26th September.

Sept 24-28

Chapter 3 : Quotients and Homomorphisms (73-111)

  • composition series and the Holder programme
  • transpositions and the alternating group

Page 106, exercises 7, 8.
Page 111, exercises 4, 5, 11, 12, 16.
Due Wednesday 3rd October.

Oct 1-5

Chapter 4 : Group Actions (112-151)

  • group actions and permutation representations
  • Cayley's Theorem
  • the Class Equation

Page 121, exercises 4, 7b, 10.
Page 130, exercises 6, 19, 28, 29.
Due Wednesday 10th October.

Oct 8-12

Chapter 4 : Group Actions (112-151)

  • automorphisms
  • Sylow's Theorem

Page 137, exercises 1, 4, 8, 12, 19.
Page 146, exercises 8, 10, 13, 17, 30.
Due Wednesday 17th October.

Oct 15-19

Chapter 5 : Direct and semidirect products, abelian groups (152-187)

  • direct products
  • recognizing direct products
  • semidirect products

No homework this week.
Attempt the practise midterm exams.

Oct 22-26

Revision for midterm exam
Midterm exam : Wednesday 6pm
Chapter 5 : Direct and semidirect products, abelian groups (152-187)

  • Fundamental Theorem of Finitely-Generated Abelian Groups

Page 156, exercise 5.
Page 165, exercises 2b, 3b, 4bc, 8.
Page 169, exercise 1 (you can omit Z_4*D_8).
Page 173, exercises 4, 17.
Page 184, exercises 8, 11.
Due Friday 2nd November.

Oct 29-Nov 2

Chapter 5 : Direct and semidirect products, abelian groups (152-187)

  • table of groups of small order
Chapter 7 : Introduction to rings (223-269)
  • definition and examples
  • polynomial rings
  • homomorphisms and quotient rings

Page 230, exercises 5, 8, 14, 24 (D=3 and 5 only), 27.
Page 247, exercises 4, 10, 12ab, 17, 34.
Due Friday 9th November.

Nov 5-9

Chapter 7 : Introduction to rings (223-269)

  • homomorphisms and quotient rings
  • ideals
  • rings of fractions
  • the Chinese Remainder Theorem

Page 256, exercises 11, 13, 14, 25, 30, 37.
Page 264, exercise 3.
Page 267, exercise 7.
Due Friday 16th November.

Nov 12-16

Chapter 8 : Euclidean domains, PIDs, and UFDs (270-294)

  • Euclidean domains
  • principal ideal domains

Page 277, exercises 3, 6, 7 ((85,1+13i) only), 9, 10.
Page 282, exercises 5a (I_2 only), 5bc, 6, 8.
Due Wednesday 28th November.

Nov 19-23

Thanksgiving break

.

Nov 26-30

Chapter 8 : Euclidean domains, PIDs, and UFDs (270-294)

  • unique factorization domains
Chapter 9 : Polynomial rings (295-335)
  • definitions and basic properties
  • polynomial rings over fields

Page 292, exercises 5, 8.
Page 298, exercises 5, 6, 13.
Page 301, exercises 3, 8.
Page 306, exercise 4.
Page 311, exercises 2cd, 9.
These exercises do not need to be handed in.

Dec 3-7

Chapter 9 : Polynomial rings (295-335)

  • polynomial rings that are UFDs
  • irreducibility criteria
Revision for final exam

.

Dec 10-14

Final exam : Monday 1:30pm

.

Back to the main page.

This page last modified by Justin Sawon
Thursday, 29-Nov-2007 14:48:09 MST
Email corrections and comments to sawon@math.colostate.edu