MATH 4020(6.0A) Algebra II, 2007/2008
Time: T/R 11:30--13:00pm
Location: MC 213
INSTRUCTOR: Yun Gao,
Office: Ross S624,
Telephone: 736-2100 ext. 33952
Web page: http://www.math.yorku.ca/Who/Faculty/YGao
OFFICE HOURS: Tuesdays and Thursdays: 13:00--14:30pm or by appointment.
John B. Fraleigh, A First Course in Abstract Algebra
7th edition, published by Addison Wesley, 2003
TOPICS TO BE COVERED:
Continuation of Algebra I (Math 3020) with applications:
groups (finitely generated Abelian groups, solvable groups, simplicity of alternating groups, group actions,
Sylow’s theorems, generators and relations);
fields (splitting fields, finite fields, Galois theory, solvability of equations).
Algebra is the study
of algebraic systems, that is, sets of elements endowed with certain
operations. A familiar example is the set of integers
with the operations of addition and multiplication. Algebra is used in almost every branch of mathematics; indeed, it has simplified the study
of mathematics by indicating connections between seemingly unrelated topics. In addition the success of the methods of algebra in unraveling
the structure of complicated systems has led to its use in many fields outside of mathematics.
This course aims to broaden and deepen the student’s knowledge and understanding of modern abstract algebra by building on the material
of MATH 3020 6.0 (or a comparable course which students may have taken). In addition to the topics listed in the Calendar, the following
may be expounded:
Group theory: Composition series.
Ring theory: General ring theory, factorization in domains.
Field theory: Ruler and compass constructions.
Most of the course material may be found in the book by John B. Fraleigh, Abstract Algebra, 7th ed. (Addison-Wesley, 2003).
AS/SC/AK/MATH 3020 6.0 or permission of the course coordinator.
SYLLABUS: We will study the
Chapter 3: Homomorphisms and Factor Groups
15, factor-group computation and simple groups
16, group action on a set
17, applications of G-sets to counting
Chapter 4: Rings and Fields
24, noncommutative examples
Chapter 5: Ideals and Factor Rings
28, Gröbner bases for ideas
Chapter 6: Extensions
31, algebraic extensions
32, geometric construction
33, Finite fields
Chapter 7: Advanced Group Theory
34, isomorphism theorems
35, series of groups
36, Sylow theorems
37, aplications of the Sylow theory
38, free abelian groups
39, free groups
40, group presentations
Chapter 9: Factorization
45, unique factirization domains
46, Euclidean domains
47, Gaussian integers and multiplicative norms
Chapter 10: Automorphisms and Galois Theory
48, Automorphisms of Fields
49, the isomorphism extension theorem
50, splitting fields
51, separable extensions
53, Galois theory
54, illustrations of Galois theory
55, cyclotomic extensions
56, insolvablility of the quintic
You are expected
to do all of the assigned homework. The only way to
learn algebra is to do it!
EXAMS: There will be four midterm tests and the final exam.
Thursday, October 4; Tuesdays, November 13, January
22, March 4.
Marks: Each of the three tests (Oct.4, Jan.22, Mar. 4 ) 15%; the test ( Nov.13) 20%; Final exam 35%.
MISSED EXAMS: There will be no
make-up exams for missed tests. Upon presentation of documentation
of a valid excuse,
the corresponding percentage of the final mark will be added to the final exam. With no presentation of such documentation
a grade of zero will be entered for the missed tests.
Drop deadline: February 1, 2008.
York Undergraduate Math Program