COURSE OUTLINE

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

           Abstract  Algebra  Online