Math 627B: Modern Algebra II
Spring 2013

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, 2006.

H is

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

A is

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

IVA IV.4
Day Topics Sections in IVA
Th. 1/17 Rings, fields.
Polynomial rings and geometry. 		IVA 1.1,2
Tu. 1/22 The polynomial ring in one variable:Main Properties IVA 1.5, H. Ch. 1-5
Th. 1/24 Ideals in polynomial rings.
Varieties: Solutions of an ideal.		 IVA 1. 2,4-5
Tu. 1/29 Homomorphisms and quotient rings. H. Ch 6, IVA, 1.4 A 2.1-2
Th. 1/31 Prime, maximal, and radical ideals
and associated quotient rings. 		H Ch. 6, A Ch. 2.3,4
Tu. 2/5 The Chinese Remainder Theorem.
Some varieties. Some parametrizations. 		IVA 1.3,4
Th. 2/7 Parametrization and implicitization.
Monomial Orderings		 IVA 1.3,4,
IVA 2.1,2
Tu. 2/12 Division in multivariate polynomial rings.
A menagerie of monomial orderings.		 IVA 2.2, 3
Th. 2/14 Dickson's Lemma. IVA 2.4
Tu. 2/19 Hilbert basis theorem, Groebner basis.
Systems of representatives and Groebner bases.		 IVA 2.5-8
Th. 2/21 S-polynomials and Grobner basis.
Buchberger's algorithm.		 IVA 2.6-8
Tu. 2/26 Modules over a ring. A Secs. 4.1,2 (just bits of each)
Th. 2/28 The rank of free modules.
Presentations of a module		 A Sec. 4.3
Tu. 3/5 Groebner basis and the syzygy module. IVA. 2.8,9
Th. 3/7 The syzygy module: an example and main theorem.
localization/rings of fractions.		 A Ch. 2.8
Tu. 3/12 More on fractions.
Euclidean domains, principal ideal domains, uniqure factorization domains.		 A Sec. 2.7,8
Th. 3/14 PID implies UFD. R a UFD implies R[x] a UFD. A. Sec. 2.6,7
Tu. 3/19 Irreducibility and Eisenstein
Resultants.		 A Sec. 2.9, IVA Sec. 5.5,6
Th. 3/21 Elimination and Extension Theorems. IVA Sec. 5.1,2
Tu. 3/26 Radical ideals and Hilbert's Nullstellensatz.
Zariski closure.		 IV.1,2
Th. 3/28 Prime ideals and irreducible varieties. Irreducible decomposition. IVA IV.5,6
Tu. 4/9 Ideal quotients.
Review before Test.
Th. 4/11 Test. See review sheet.
Tu. 4/16 Parametrizations and implicitization. I.3, III.3, V.1.
Th. 4/18 Computing in the coordinate ring of a variety.
Finite varieties. 		V.3
Tu. 4/23 Cone varieties, homogeneous ideals, the cone over a variety. VIII.1-4x.
Th. 4/25 Projective geometry. VIII.1-4.
Tu. 4/30 No class: discuss projects with me. .
Th. 5/2 No class: discuss projects with me. .
Tu. 5/7 Presentations. .
Tu. 5/9 Presentations. .