Grad Algebra II

Textbook: None.
Course Notes Date Details
Homework 1 Solutions Tues, Feb 4 Course Notes
Abstract Galois Connections
Field of Fractions
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])
Final Exam
Thurs, May 1
Algebra Qual
Fri, June 6 Review Session 1
Review Session 2
Review Session 3
Review Session 4