SYLLABUS

Course:       MATH 405  Abstract Algebra             Term:   Winter 2007-08             Meets:  MWF 8:00 - 9:10

Instructor:   Craig Kalicki        Office:  H-284       E-mail:  craig.kalicki@briarcliff.edu       Phone:  279-5541

 

Text:            Nicodemi, Sutherland, Towsley, An Introduction to Abstract Algebra, First Edition, Pearson
                       Prentice Hall, ã2007  ISBN 0-13-101963-5

 

Goals of course:

 

This is a course in abstract mathematics.  Its purpose is to survey some of the fundamental structures of

algebra using an approach that promotes investigation and discovery, abstraction and generalization, and

logical argumentation.  Students in this course should expect to

    1)  gain an appreciation of the nature of algebra as a field of mathematical study.

    2)  acquire a knowledge base of fundamental concepts and results of algebra.

    3)  develop skills in the processes of inductive reasoning, abstraction, and generalization.

    4)  develop skills in constructing and communicating deductive arguments in the formal language of

             mathematics.

For those intending to teach mathematics at the secondary level, this course provides a firm foundation of

knowledge and algebraic skills essential for effective teaching.

 

Prerequisite:   MATH 245

 

Expectations of students:

 

1)  Attendance at all class meetings is expected.

2)  Read each section of the text casually; then, go back to examine some of the details.  Pay particular

        attention to definitions of terms and examples.

3)  Most of the theory in this course is available in the textbook.  You will not need to take extensive notes;

        lecture notes will be posted online as the material is covered.  The most important activities during

        class are listening and asking questions.

4)  Assignments will be accepted only on the posted due dates.  No late work will be accepted except in

        cases of serious illness or family emergency. 

5)  Time constraints do not permit solution of all problems in class.  Make use of your instructor’s office

        hours for help with assigned problems.  Working with others in the course is highly recommended.

        Selected problem solutions will be placed on reserve in the main library.

6)  Show evidence of learning on exams.  Exams will not be given during class periods; exam times will

        be arranged individually.  Exam questions will involve definitions of terms, illustrative examples,

        and proofs involving basic concepts.

 

Grading: 

Two exams @ 25%

50%

 

A   = 100 - 91

C+ = 71 - 68

Final exam

30%

 

A-  =   90 - 87

C  =  67 - 61

Assignments

20%

 

B+ =   86 - 83

D+ = 60 - 57

 

 

 

B   =   82 - 76

D  =  56 - 50

Total

100%

 

B-  =   75 - 72

 

 

Grading rubric:

 

The following rubric will be used to assess written work including exam questions and assigned problems.

 

         Level                                Characteristics

 

   (4)  Superior            Shows superior knowledge of the material

                                 Selects appropriate mathematical tools and applies them correctly

                                 Presents complete and readable solutions of problems

                                 Shows superior awareness of interrelationships among concepts

                                 Writes and communicates concise, coherent, valid arguments                   

 

   (3)  Good                Shows above average knowledge of the material

                                 Selects appropriate mathematical tools and applies them correctly

                                 Presents generally readable and complete solutions of problems

                                 Shows high level of awareness of interrelationships among concepts

                                 Writes and communicates generally coherent and valid arguments

 

   (2)  Adequate          Shows adequate knowledge of the material

                                 Selects appropriate mathematical tools and applies them with minor errors

                                 Presents acceptable solutions of problems with occasional errors or omissions

                                 Has some awareness of interrelationships among concepts

                                 Writes and communicates minimally acceptable arguments with occasional errors

 

   (1)  Minimal             Shows minimally acceptable knowledge of the material

                                 Selects appropriate mathematical tools and applies them incorrectly or

                                    selects inappropriate tools

                                 Presents minimally acceptable solutions of problems with errors or omissions

                                 Has little awareness of interrelationships among concepts

                                 Shows little ability to write and communicate coherent arguments

 

   (0)  Unacceptable   Has inadequate knowledge of the material

                                 Chooses inappropriate mathematical tools or fails to apply tools correctly

                                 Is unable to solve most problems

                                 Is generally unaware of interrelationships among concepts

                                 Is unable to write and communicate coherent arguments

 

Student learning outcomes:

 

Expected learning outcomes specific to this course are listed below.  Assessment of the degrees to which

these outcomes have been achieved will be done via written exams and graded homework assignments.

 

Students who have complete MATH 405 will be able to

1)   demonstrate understanding and verify basic algebraic results about the integers.

2)   perform and apply modular arithmetic.

3)   define and verify basic results about the algebraic structures of ring, integral domain, and field.

4)   demonstrate understanding of the field of complex numbers.

5)   verify and apply basic results about polynomials over a commutative ring.

6)   define the concept of group and verify basic results about the group structure.

7)   supply examples and verify basic results about abelian groups, cyclic groups, and permutation groups.

8)   demonstrate understanding of the concept of isomorphism of groups.

9)   construct examples and verify basic results about quotient groups and quotient rings.

10) demonstrate understanding of the concept of algebraic extension of a field. 

 

Text material:

 

Chapter 1   Topics in Number Theory  [6 classes]

   Well-ordering principle /  mathematical induction /  equivalence relations /  division algorithm /  gcd and

   lcd /  Euclidean algorithm /  primes /  Fundamental Theorem of Arithmetic / Theorems of Euler and Fermat.   

 

Chapter 2   Modular Arithmetic and Systems of Numbers  [6 classes]

   Congruence mod m /  divisibility tests /  ring of integers mod m /  rings / integral domains /  fields /

   field of complex numbers.

 

Chapter 3   Polynomials  [5 classes]

   Polynomials over a commutative ring /  division algorithm for polynomials /  Remainder and Factor

   Theorems / irreducibility /  unique factorization /  Fundamental Theorem of Algebra /  polynomial

   congruences.

 

Chapter 4   A First Look at Group Theory  [7 classes]

   Groups /  groups of symmetries /  abelian groups /  subgroups /  direct products /  homomorphism and

   isomorphism /  cyclic groups /  permutation groups / Cayley’s Theorem.
    

Chapter 5   New Structures from Old  [3 classes]

   Cosets /  Lagrange’s Theorem /  normal subgroups / quotient groups /  subrings /  ideals /  quotient

   rings.

 

Chapter 6   Looking Forward and Backward  [2 classes]

   Extension fields /  algebraic extensions /  constructible numbers. 

 

Ó2007 by Craig Kalicki PhD, Briar Cliff University, Sioux City, IA  51104