Fall 2023 Algebra
上課時間: 週三67，週五678.
上課地點: 天數101
Office hour: 週三3:30--5:30 at 天數412
The purpose of this course is to introduce basic tools
from algebra in the research of various branches of mathematics.
The goal is to familiarize students with the use of these
algebraic tools in problem solving.
References:
Abstract Algebra,
3rd edition by I.N. Herstein
Fields and Galois Theory, by J.S. Milne
Additional reference:
Finite groups: An
Introduction, J.-P. Serre (arXiv)
Linear Representations
of finite groups, J.-P. Serre
(GTM42)
Abstract Algebra, Dummit
and Foote
Theory of commutative fields, M. Nagata
Handouts and homework will published in NTU Cool every week.
Contents in handouts are updated frequently.
Syllabus:
Group theory:
Week 1: Motivation and
examples. Basic
definitions in group theory. Group
homomorphisms. [CH.2, SEC 1-4]
Week 2: Normal subgroups and
quotient groups. Group actions and the orbit formula. [CH.2, SEC
5-9]
Week
3: Burside's lemma. Sylow theorem. Jordan-Holder
Theorem. Solvable groups. [CH.2,
SEC 11], [CH1-2]
Week 4: Examples of simple groups
and structure of finite abelian groups. [CH.3],
[CH.6, SEC1]
Week 5: Basic representation theory
of finite groups. [Part I,
p.1-43], [Chapter
8, 8.1-8.4]
Week 6: Burnside's Theorem. [Chapter
8, 8.6, 8.10,
8.11]
Ring
theory (I):
Week 7: Ideals,
quotients and
modules. Examples:
quaternion algebras
and polynomial
rings.
Week 8:
Midterm
Week
9: Finitely
generated modules
over PID.
Fields and
Galois theory:
Week 10: Fields extensions.
Algebraic and normal extensions.
Week 11: Algebraic closures. Separable
extensions and Galois extensions.
Week 12: Galois correspondence.
Week 13: The
straight-edge and compass
construction. Solvability
of polynomial
equations.
Week 14: Computing
Galois group by reduction modulo a
prime. Kummer Theory.
Week
15: Infinite Galois theory.
習
題課時間: 每週五第八節
助教: 李俊緯
(d09221001AT ntu.edu.tw)
作業: 繳交每週
發佈於NTU
Cool的作業。
建
議自行做Herstein書中進度對應章節的
Problems (Harder和Very Hard除外)。
評量方式:
作業 (30%) 週五第八節結束前繳交，請利用NTU COOL上傳作業， 逾期不收。
期中考(35%) October 27,
13:20--15:10 (Week 1-6)
期末考(35%) December 22, 13:20--16:30 (Week 7-14)