Course Syllabus

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Instructor:

Prof. Ivan Chi Ho IP
Email: ivan.ip@ust.hk
Office: Room 3483 (Lift 25-26)
Office Hour: by appointment

Class Photo [Image1] [Image2]

Class Hour and Venue:

Monday 13:30 - 16:20 @ Cheng Yu Tung Building (CYT) G003

 

Course Description:

In this course, we will study the fundamentals of the theory of groups, rings, modules and fields. The goal of this course is to prove several important structural theorems for these abstract algebraic objects, including the Sylow's Theorem for finite groups, the Chinese Remainder Theorem, the Fundamental Theorem for finitely generated modules over P.I.D., as well as the Fundamental Theorems of Galois Theory. We will also introduce category theory, which provides a general framework for these concepts.

 

Prerequisite:

None*. We will start everything from scratch, but with a pace much faster than a usual undergraduate course, so some mathematical maturity is required. Straightforward proofs are usually left as exercises.

*I assume you know what is LaTeX: \mathbb{Z}, \mathbb{Q}, \mathbb{R}, \mathbb{C} and some elementary set theory.

References:

Lecture Notes (Modules)

[Main] Abstract Algebra (3rd Edition) by Dummit & Foote [here] (on Library Reserve)

Algebra (Revised 3rd Edition) by S. Lang [here] (on Library Reserve)

Abstract Algebra: The Basic Graduate Year by R. Ash [here]

 

Grading:

There will be 4 take-home examinations, one for each topic on groups, rings, modules and fields. They weigh 25% each and will be announced 1 week before the due date. You are required to complete them by LaTeX with the template provided.

 

LaTeX:

Since this is a PG course, we encourage every student to become familiar with LaTeX, the basic typesetting language commonly used in mathematical research. Follow the instruction here to install a full TeX distribution according to your OS. Then download one of the typesetting software. Personally, I recommend TexWorks, but TexStudio or TexMaker are also popular. Try compiling this template to make sure you are able to run LaTeX correctly.

 

Tentative Schedule:

(Week 1 - 3) Group Theory

(Week 4) Category Theory

(Week 5 - 7) Ring Theory

(Week 8 - 10) Module Theory

(Week 11 - 13) Field Theory

 

Class Log:

Date Topics References
2018-09-03

Introduction

Group Theory
Fundamentals of Groups, Generators and Relations, Cyclic Groups, Cosets, Quotient Groups, Lagrange's Theorem, Isomorphism Theorems x 4, Group Actions, Orbit Stabilizer Theorem, Cayley's Theorem.

Dummit & Foote
Ch 1.1-1.7
Ch 2.1-2.6
Ch 3.1-3.3
Ch 4.1-4.2
2018-09-10

Conjugacy Classes, Simplicity of LaTeX: A_n, N/C Theorem, Characteristic Subroups, Properties of LaTeX: p-Groups, Cauchy's Theorem, Sylow's Theorem, Fundamental Theorems of Finitely Generated Abelian Groups (without proof).

Simplicity of Groups of Order: LaTeX: 12, 30, 56, 60, 105, LaTeX: pq, LaTeX: p^2q

 Dummit & Foote
Ch 3.5
Ch 4.3-4.6
Ch 5.1-5.2
Ch 6.1
2018-09-17 CLASS CANCELLED due to Typhoon Mangkhut  
2018-09-24

Automorphisms of LaTeX: Z_n, Semidirect Product, Composition Series, Jordan-Hölder Theorem, Upper/Lower Central Series, Derived Series, Nilpotent Groups, Solvable Groups, Free Groups. Simple Groups of Order 168.

Classification of Groups of Order: LaTeX: 12, 30, 56, pq.

 Dummit & Foote
Ch 3.4
Ch 5.4-5.5
Ch 6.1-6.3
2018-10-01 NO CLASS  
2018-10-08

Category Theory

Category, Functors, Natural Transformations, Initial Object, Terminal Object, Universal Properties, Product & Coproduct, Fiber Product & Coproduct, Cones, Direct Limit, Inverse Limit, Adjoint Functors, Representable Functors, Yoneda's Lemma.

Special Categories and Constructions: Discrete Category, Finite Category, Small Category, Abelian Category, Groupoids, Opposite Category, Slice Category, Comma Category

Examples: Set, Monoid, Group, Ab, Ring, CRing, R-Mod, Vect, G, G-Set, Top, Cat

Dummit & Foote
Appendix II

Lang
Ch I.11, III.10

Wikipedia

 
2018-10-15

Ring Theory

Fundamental of Rings, Division Rings and Fields, Ideals, Quotient Rings, Maximal and Prime Ideals, Zorn's Lemma (without proof), Inverse Limits, Ring of Fractions, Local Rings, Chinese Remainder Theorem.

Examples: Matrix Rings, Group Rings

Dummit & Foote
Ch 7.1-7.6
Ch 15.4
Appendix I
2018-10-22

Quadratic Integer Rings, Euclidean Domain (Euclidean Algorithm, Universal Side Divisors), Principal Ideal Domains (Dedekind-Hasse Norm), Unique Factorization Domains, Polynomial Rings (Gauss' Lemma, Eisenstein's Criterion), Noetherian Rings.

Examples: LaTeX: \mathbb{Z}[i], \mathbb{Z}[\sqrt{-5}], \mathbb{Z}[\frac{1+\sqrt{-19}}{2}], \mathbb{Z}[x]

Dummit & Foote
Ch 8.1-8.3
Ch 9.1-9.5
Ch 15.1
2018-10-29 Hilbert Basis Theorem, Hilbert Nullstellensatz (without proof), Radical, Primary Decomposition Theorem, Artinian Rings, Discrete Valuation Ring, Integrally Closed Domain, Fractional Ideals, Dedekind Domains

Dummit & Foote
Ch 15.1-15.3
Ch 16.1-16.3
2018-11-05

Module Theory

Fundamentals of Modules, Quotient Modules, R-Algebras, Nakayama's Lemma, Direct Sum, Free Modules, Tensor Products of Rings / Bimodules / R-Algebras, Tensor Algebras, Symmetric Algebras, Exterior Algebras.

 Dummit & Foote
Ch 10.1-10.4
Ch 11.5
2018-11-12

Proof of Fundamental Theorem of Finitely Generated Modules over P.I.D., Rational and Jordan Canonical Form.

Introduction to Homological Algebra

Short Exact Sequences, Splitting Sequences, Snake Lemma, Exact Functors, Hom Functors, Projective Modules.

 Dummit & Foote
Ch 12.1-12.3
Ch 10.5
2018-11-19

Injective Modules, Flat Modules, Adjoint Associativity.

Homological Algebra, Short to Long Exact Sequence, Ext Functor, Tor Functor, Balanced Bifunctor, Derived Functors.

 Dummit & Foote
Ch 10.5
Ch 17.1
2018-11-26

Field Theory

Fundamentals of Fields, Algebraic Extension, Composite Field, Splitting Field, Algebraic Closure, Separable Extension, Cyclotomic Extension, Finite Fields

Ruler and Compass Construction.

Dummit & Foote
Ch 13.1-13.6
2018-12-03

Galois Theory

Automorphism Groups, Galois Extension, Fundamental Theorems of Galois Theory, Linear Independence of Characters, Galois Theory over Finite Fields, Composite Extensions, Primitive Element Theorem

 Dummit & Foote
Ch 14.1-14.4
2018-12-06
(Rm 3494)		 Cyclic and Abelian Extensions, Solvability of General Polynomials, Galois Groups of Polynomials of Low Degree, Inseparable Extensions, Field Norm and Trace, Hilbert's Theorem 90, Group Cohomology, Kummer Theory, Infinite Galois Theory

Dummit & Foote
Ch 14.5-14.9,
Ch 17.2-17.3

Lang
Ch V.3, V.4, V.6
Ch VI.5, VI.8, VI.10, VI.14

Notes by K. Conrad (Norm and Trace)

Notes by K. Harper (Kummer Theory)

 

 

Course Summary:

Date Details Due