# Grad Algebra II

 Course Notes Date Details Textbook: None. 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]) Final Exam Solutions Thurs, May 1 Algebra Qual Fri, June 6 Review Session 1 Review Session 2 Review Session 3 Review Session 4