Math 627A: Modern Algebra I
Spring 2011

Instr: Mike O'Sullivan


Schedule

I will try to keep to this schedule but will update it as needed.

IVA is

Cox, Little, O'Shea Ideals, Varieties, and Algorithms:
An Introduction to Computational Algebraic Geometry and Commutative Algebra
2nd or 3rd Ed., Springer-Verlag, 1997, 200.

H is

Hungerford Abstract Algebra: An Introduction, Hungerford. 2nd Ed., Harcourt, 1997.

A is

Ash, Abstract Algebra: The basic graduate year , online.

Day Topics Sections in IVA
Mon. 8/29 The integers and the polynomial ring in one variable:
Main Theorems 		IVA 1.5, H. Ch. 1-5
Wed. 8/31 The computer algebra system Sage. IVA 1.5, H. Ch. 1-5
Wed. 9/7 Ideals in polynomial rings.
Varieties: Solutions of an ideal.		 IVA 1.1-2, 4-5
Mon. 9/12 Parametrization and Implicitization.
 I.2,3. II.1
Wed. 9/14 Summary: Ideal of a set of points;
Variety of a set of polynomials.		 Ch. 1
Mon. 9/19 Monomial Orderings
Division in multivariate polynomial rings.		 II.2, 3
Wed. 9/21 Division in multivariate polynomial rings.
A menagerie of monomial orderings.		 II.2, 3
Mon. 9/26 Dickson's Lemma. II.4
Wed. 9/28 Hilbert basis theorem, Grobner basis. II.5-8
Mon. 10/3 Rings, ideals.
Homomorphisms, quotient rings.
Systems of representatives and Groebner bases.		 H Ch. 6
A Ch. 2.1-2
Wed. 10/5 Quotient rings, and the isomorphism theorems.
prime and maximal ideals.		 H Ch. 6
A Ch. 2.3-4
Mon. 10/10 Prime, radical, and maximal ideals
and quotient rings.		 A Sec. 2.4
Wed. 10/12 Injective homomorphisms and
localization/rings of fractions.		 A Ch. 2.8
Mon. 10/17 Modules over a ring. A Secs. 4.1,2 (just bits of each)
Wed. 10/19 The rank of free modules.
Presentations of a module		 A Sec. 4.3
Mon. 10/22 Principal ideal domains, uniqure factorization domains;
primes and irreducibles.		 A Sec. 2.6
Wed. 10/24 Classification of localizations of Z . A. Sec. 2.8
Mon. 10/31 Unique factorization domains.
PID implies UFD.		 A Sec. 2.9, IVA Sec. 5.5,6
Wed. 11/2 R a UFD implies R[x]. A Sec. 2 and my notes
Mon. 11/7 Generators and relations for a module.
Syzygy module for a monomial ideal.		 A Sec. 4.3, IVA II.6-9, my notes.
Wed. 11/9 Review before exam. .
Mon. 11/14 Test. See review sheet.
Wed. 11/16 Syzygy module for a Groebner basis.
Elimination and Extension Theorems.		 III.1,2
Mon. 11/21 Extension and Closure Theorems.
Implicitization.		 IVA III.2,3
Mon. 11/28 Radical ideals and Hilbert's Nullstellensatz.
Zariski closure.		 IV.1,2
Wed. 11/30 The algebra-geometry dictionary.
Sums, products and intersections of ideals.
Computation of the intersection of two ideals. 		IV.2,3.
Mon. 12/5 Proof of the closure theorem. Ideal quotients, and the geometric interpretation. IV.3,4.
Wed. 12/7 Irreducible varieties and minimal decomposition. IV.5,6
Fri. 12/16 Final Exam. See review sheet.