All reading references refer to Dummit and Foote's Abstract Algebra, 3rd edition, except when explicitly marked otherwise.

For more of a progression through the book and videos at the undergraduate level, please visit Taylor's excellent list of videos on abstract algebra

An excellent graduate course on group theory, taught by Richard Borcherds

An excellent graduate course on Galois theory, also taught by Richard Borcherds

Syllabus for Algebra qualifying exam

Jump to: Week 2 · Week 3 · Week 4 · Week 5 · Week 6 · Week 7 · Week 9 · Week 10 · Week 11 · Week 12 · Week 13

Week 1

Reading

On using LaTeX:

Videos

On using LaTeX:

On the dihedral group:

Week 2

Reading

Dummit and Foote reading:

  • Chapter 2

Further reading for qual problems:

  • Problem 1: Section 4.1
  • Problem 2: Sections 3.3 and 4.4
  • Problem 4: Problem 22 of Section 1.6, Section 4.4

Videos

Week 3

Reading

Dummit and Foote reading:

  • Chapter 3

Further reading for qual problems:

  • Problems 1, 3, 4: Section 6.1

Videos

On group homomorphisms:

On the isomorphism theorems:

On solvable groups:

On direct products:

Week 4

Reading

Dummit and Foote reading:

  • Sections 4.1, 4.2, 4.3, 4.4 and 4.6
    (all of Chapter 4 except Section 4.5)

Further reading for qual problems:

Videos

On group actions:

Week 5

Reading

Dummit and Foote reading:

  • Section 4.5

Videos

On the Sylow theorems:

Week 6

Reading

Dummit and Foote reading:

  • Chapter 5

Further reading for qual problems:

  • Section 1.4: Matrix groups (#3)
  • Section 6.3: A word on free groups (#4)

Videos

On direct products:

On semidirect products:

Week 7

Reading

Dummit and Foote reading:

  • Chapter 6

Videos

On p-groups:

On nilpotent groups:

On groups of medium order:

On free groups:

Week 9

Reading

Dummit and Foote reading:

  • Sections 13.1, 13.2, 13.4, 13.5

Videos

From Borcherd's course on Galois theory:

Week 10

Reading

Dummit and Foote reading:

  • Sections 14.1, 14.2, 14.4

Videos

From Borcherd's course on Galois theory:

Week 11

Reading

Dummit and Foote reading:

  • Sections 14.6, 14.7, 14.8, 14.9

Videos

From Borcherd's course on Galois theory:

Week 12

Reading

Topic:

  • Finite fields

Dummit and Foote reading:

  • Sections 13.5, 14.3
  • Section 13.1: Example 6
  • Section 13.2: Example (Quadratic Extensions over Fields of Characteristic ≠ 2)
  • Section 14.1: Examples 2, 7 (finite extensions of finite fields are Galois), 8
  • Section 14.4: Example 2

Videos

From Borcherd's course on Galois theory:

Week 13

Reading

Topic:

  • Cyclotomic extensions

Dummit and Foote reading:

  • Sections 13.6, 14.5
  • Section 13.4: Example (Splitting field of x^n-1: Cyclotomic Fields), Example (Splitting Field of x^p-2, p a prime)
  • Section 14.2: Example (Splitting Field of x^8-2)

Videos

From Borcherd's course on Galois theory: