**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

e-mail:
ygao@yorku.ca

Web page:
http://www.math.yorku.ca/Who/Faculty/YGao

OFFICE HOURS: Tuesdays and Thursdays: 13:00--14:30pm or by appointment.

TEXTBOOK:

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).

**Prerequisite**:
AS/SC/AK/MATH 3020 6.0 or permission of the course coordinator.

SYLLABUS: We will study the
following sections:

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

HOMEWORK:
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.

Tests:
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.

IMPORTANT DATES:

Drop deadline: February 1, 2008.

HELPFUL LINKS:

York Undergraduate
Math Program