Grad Algebra II
| Textbook: None. | ||
| Course Notes | Date | Details |
|---|---|---|
| Homework 1 Solutions | Tues, Feb 4 |
Course Notes Abstract Galois Connections Field of Fractions Localization C(X) for Compact, Hausdorff X Weak Nullstellensatz for C(X) Localization = Germs of Functions |
| Homework 2 Solutions | Tues, Feb 25 |
Course Notes What do Z and K[x] have in common? Euclidean => PID => UFD Noetherian Rings Who cares about unique factorization? Riemann Surfaces vs. Algebraic Integers |
| Homework 3 Solutions | Thurs, Mar 20 |
Gauss' Lemma (R UFD implies R[y] UFD) Z[y] and K[x,y] are UFDs, but they are not PIDs. Nonmaximal primes in Z[y] and K[x,y] |
| Homework 4 Solutions | Thurs , Apr 17 |
Course Notes Ring Extensions and Evaluation Morphisms K-Algebras vs. K-Modules F.G. & Algebraic => Field (not hard) F.G. & Field => Algebraic (hard) K[x1,..,xn] is a UFD (Gauss' Lemma) K[x1,..,xn] is Noetherian (Hilbert's Basis Theorem) Weak Nullstellensatz |
| No HW5 |
Course Notes Variety = Radical Ideal (Strong NSS) Irreducible Variety = Prime Ideal Krull Dimension Algebraic Independence Transcendence Degree Noether Normalization Zariski's Lemma Proof of the NSS Epilogue: dim(V) = tr.deg(K[V]) |
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Final Exam Solutions |
Thurs, May 1 | |
| Algebra Qual |
Fri, June 6 |
Review Session 1 Review Session 2 Review Session 3 Review Session 4 |