MATH5111 (L1) - Advanced Algebra I

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Instructor:
Instructor Prof. Ivan Chi Ho IP
Email:  ivan.ip@ust.hk
Office:  Room 3470 (Lift 25-26)
Office Hour:  By appointment

Class Hour and Venue:

Wednesday 15:00 - 17:50 @ Cheng Yu Tung Building (CYT) G001

ClassPhoto1 ClassPhoto2

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

  • Sylow's Theorem for Finite Groups,
  • Chinese Remainder Theorem,
  • Structure Theorem for Finitely Generated Modules over a P.I.D.,
  • Theory of Homological Algebra, as well as
  • Fundamental Theorems of Galois Theory.

We will also emphasize the language of category theory, which provides a general framework for these concepts.

 

Prerequisites:

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 may be left as exercises.

*I assume you know what are LaTeX: \mathbb{Z}, \mathbb{Q}, \mathbb{R}, \mathbb{C} and know some elementary set theory.
**MATH3131 is fully contained in this course and is not a prerequisite. If you have taken 3131, please expect some overlaps of materials. On the other hand, MATH3121 is about 1/100 of the course materials covered here.

References:

Instructor's Lecture Notes 

[Main] Abstract Algebra (3rd Edition) by Dummit & Foote [here] (Roughly Chapter 1-17)

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

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, each with one additional question related to category theory. They weigh 25% each. You are required to complete them by LaTeX with the template provided within 48 hours.

 

Tentative Schedule:

(Lecture 1 - 3) Group Theory

(Lecture 4) Category Theory

(Lecture 5 - 7) Ring Theory

(Lecture 8 - 10) Module Theory

(Lecture 11 - 13) Field Theory

 

Class Log:

Date Topic References

Lecture #01

2023-09-06

Introduction

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

 Dummit & Foote
Ch 1.1-1.7
Ch 2.1-2.6
Ch 3.1-3.3

Lecture #02

2023-09-13

Group Actions, Orbit Stabilizer Theorem, Cayley's Theorem, Conjugacy Classes, Simplicity of LaTeX: A_n, N/C Theorem, Characteristic Subgroups, Properties of LaTeX: p-Groups, Cauchy's Theorem, Sylow's Theorem

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

 Dummit & Foote
Ch 3.5
Ch 4.1-4.6
Ch 6.1

Lecture #03

2023-09-20

 

Fundamental Theorems of Finitely Generated Abelian Groups (without proof). Automorphisms of LaTeX: Z_n
, Semidirect Product, Composition Series, Jordan-Hölder Theorem, Upper/Lower Central Series, Derived Series, Nilpotent Groups, Solvable Groups, Free Groups.

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

 Dummit & Foote
Ch 3.4
Ch 5.1-5.5
Ch 6.1-6.3

Lecture #04

2023-09-27

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

Dummit & Foote
Appendix II

Lang
Ch I.11, III.10

Wikipedia

 

Lecture #05

2023-10-04

 

Ring Theory

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

Examples: Matrix Rings, Group Rings

 

Dummit & Foote
Ch 7.1-7.6
Ch 15.4
Appendix I

Lecture #06

2023-10-11

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)

Examples: LaTeX: \mathbb{Z}\left\lbrack i\right\rbrack, LaTeX: \mathbb{Z}\left\lbrack\sqrt{-5}\right\rbrack, LaTeX: \mathbb{Z}\left\lbrack\frac{1\pm\sqrt{-19}}{2}\right\rbrack, LaTeX: \mathbb{Z}\left\lbrack x\right\rbrack

Dummit & Foote
Ch 8.1-8.3
Ch 9.1-9.5

Lecture #07

2023-10-18

Noetherian Rings, Hilbert Basis Theorem, Hilbert Nullstellensatz (without proof), Radical, Primary Decomposition Theorem, Krull Dimension, Nakayama's Lemma, Jacobson Ideal, Artinian Rings, Discrete Valuation Rings, Integrally Closed Domain, Fractional Ideals, Dedekind Domains, Unique Factorization of Ideas.

Examples:  LaTeX: \mathcal{O}_K is D.D. (without proof)

Dummit & Foote
Ch 15.1-15.3
Ch 16.1-16.3

Lecture #08

2023-10-25

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

Lecture #09

2023-11-01

 

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.

 Dummit & Foote
Ch 12.1-12.3
Ch 10.5

Lecture #10

2023-11-08

 

Projective Modules, Injective Modules, Flat Modules, Adjoint Associativity.

Homological Algebra, Short to Long Exact Sequence, Ext Functor, Tor Functor.

 Dummit & Foote
Ch 10.5
Ch 17.1

Lecture #11

2023-11-15

Field Theory

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

Dummit & Foote
Ch 13.1-13.6

Lecture #12

2023-11-22

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

Lecture #13

2023-11-29

 Cyclic and Abelian Extensions, Solvability of General Polynomials, Galois Groups of Polynomials of Low Degree, 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