Math 81/111 Abstract Algebra

The official syllabus in pdf form.

Require textbook (referred to as DF):

  • David S. Dummit and Richard M. Foote, Abstract Algebra, 3rd Edition

List of other useful texts and resources:

  • J. S. Milne, Fields and Galois Theory, (referred to as FT) available online via Milne's website
  • Juliusz BrzeziƄski, Galois theory through exercises, available on-line via SpringerLink
  • Keith Conrad's expository notes
  • Serge Lang, Algebra, Graduate Texts in Mathematics, vol. 211, Third edition, 2005.
  • Ian Stewart, Galois Theory, Third edition, 2003.

Weekly problem sets will be due in class on Friday.

Weekly Syllabus and Homework

Updated March 04, 2020.

Week Date Topics Reading Homework
1 Mon 06 Jan History of solving polynomial equations. The complex numbers.
Wed 08 Jan Review of ring theory: Euclidean domains, PID, UFD. Irreducible polynomials and ideals in polynomial rings. DF 9.2
FT 7-11
Fri 10 Jan Roots. Fundamental Theorem of Arithmetic. Reduction mod p. Irreducibility criteria for polynomials. Irreducible polynomials over finite fields. Eisenstein's criterion. DF 9.2-9.5
FT 11-13
2 Mon 13 Jan Gauss's Lemma and primitive polynomials. Tower law for degrees. DF 9.3, 13.1
FT 13-17
Problem Set #1
Wed 15 Jan Finitely generated extensions. Simple extensions. Classification of simple extensions. Transcendental and algebraic elements. Minimal polynomial. DF 13.2
FT 15-19
Fri 17 Jan Algebraic extensions, continued. Compass and straightedge. Constructible numbers. Pythagorean closure. DF 13.3
FT 21-23
3 Mon 20 Jan No class: Martin Luther King, Jr. Day Problem Set #2
Wed 22 Jan Construction impossibility proofs, continued. Regular n-gons. Splitting fields. DF 13.3-4
FT 28-30
Fri 24 Jan Splitting fields. DF 13.4
FT 24-25
4 Mon 27 Jan Separability. DF 13.5
FT 30-33
Problem Set #3
Wed 29 Jan Multiple roots. Embeddings. DF 13.5
FT 30-37
Thu 30 Jan X-hour lecture: Algebraic closure. DF 13.4
FT 24-25
Fri 31 Jan Field automorphisms. Automorphism group. DF 14.1
FT 36-39
5 Mon 03 Feb Constructing automorphisms. Automorphism group calculations. Fixed fields. DF 14.1
FT 36-39
Problem Set #4
Wed 05 Feb Galois extensions. Linear independence of embeddings. DF 14.1-14.2
FT 36-37
Fri 07 Feb Fundamental theorem of Galois theory. Examples of the Galois correspondence. DF 14.2
FT 36-39
6 Mon 10 Feb End of proof of the Galois correspondence. DF 14.2
FT 38-40
Problem Set #5
Wed 12 Feb Normality. Finite fields. Cyclotomic fields. DF 14.2-3
FT 37-48
Fri 14 Feb Examples of Galois extensions. Computing Galois groups. DF 14.2
FT 52-54
7 Mon 17 Feb Applications of the Galois correspondence. Radical extensions. Solvability by radicals. FT 4
pp. 42-45
Takehome Midterm Exam
Wed 19 Feb Radical extensions. Solvability by radicals. Galois's solvability theorem. DF 14.7
FT 42-45, 74-75
Fri 20 Feb Solvability, continued. Primitive element theorem. DF 14.7
FT 59-61, 74-75
8 Mon 23 Feb Discriminant. Galois perspective on quadratic and cubic extensions. Quartic extensions. DF 14.7
FT 49-51
Problem Set #6
Wed 25 Feb Quartic extensions. DF 14.7
FT 50-58
9 Mon 02 Mar Finite fields. Cyclotomic fields. DF 14.5
FT 62-65
Problem Set #7
Wed 04 Mar Infinite Galois theory. Krull topology. Profinite groups. DF 650-652
FT 91-96
Thu 05 Mar Fundamental Theorem revisited. DF 650-652
FT 91-96
Fri 06 Mar Taussky-Todd's theorem. Grace Hopper's thesis. Notices article
10 Mon 09 Mar Final Exam! Final Exam Review



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