The contents will be adjusted as the course progresses.
- I. Group Theory
- Aug 29. The isomorphism theorems. Dedekind's modular law.
- Sep 3. Zassenhaus' Butterfly Lemma. Subnormal series, Schreier's Refinement Theorem. Simple groups.
- Sep 5. Composition series, Jordan-Hölder Theorem. Solvable groups, derived series. Nilpotent groups, lower central series.
- Sep 10. Group actions. Fixed point lemma.
- Sep 12. p-groups. Sylow theorems.
- Sep 17. Direct products. More on nilpotent groups. Lagrange converse.
- Sep 19. More on solvable groups. Hall's theorem.
- Sep 24. Simple groups. Iwasawa's Lemma. Alternating groups.
- Sep 26. Projective special linear groups.
- Oct 1. Projective geometries.
- Oct 3. Words and free monoids. Free groups.
- Oct 8. More on free groups. Presentations.
- Oct 10. Well-ordered sets, transfinite induction. Zorn's Lemma and applications.
- Oct 15. No class (Fall break).
- Oct 17. In-class midterm exam
- II. Rings and modules
- Oct 22. Rings. Maximal ideals. Chinese remainder theorem.
- Oct 24. CRT, continued. Noetherian rings. Hilbert's basis theorem.
- Oct 29. Modules. Direct sums, direct products.
- Oct 31. Free modules. Noetherian modules. Tensor products.
- Nov 5. Divisibility in integral domains. PIDs and UFDs. Euclidean domains.
- Nov 7. Quadratic rings and fields. Polynomial rings over a field and over a UFD. Gauss's Lemma.
- Nov 12. Polynomial rings over a UFD (continued). Rank of modules over integral domains.
- Nov 14. Stacked bases theorem. Structure theorems for finitely generated modules over a PID.
- III. Field extensions and introduction to algebraic geometry
- Nov 19. Field extensions. Simple extensions. Algebraic elements.
- Nov 21. Finite, algebraic and finitely generated extensions.
- Nov 26. Splitting fields. Separability.
- Nov 28. No class (Thanksgiving).
- Dec 3. Primitive element theorem. Algebraic independence and transcendence degree.
- Dec 5. Matroids. Finitely generated algebras.
- Dec 10. Zariski's Theorem. Weak Nullstellensatz. Maximal spectrum. Algebraic sets. Strong Nullstellensatz.
Final Exam due Dec 18.
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