1.       Course Description

This is an introductory course in modern algebra in mathematics education. It provides axiomatic foundation for further study of mathematics. The algebraic structures dealt in this course are groups, rings, fields and field extensions.  

2.       General Objectives

The general objectives of this course are as follows:

·         To familiarize the students with the understanding of the basic algebraic structures.

·         To develop capabilities among the students in proving theorems and problem solving techniques in algebra.

·         To help them develop positive attitude towards modern algebra.

·         To help them develop the knowledge of field extensions and to develop capabilities among the students in proving theorems of field extensions.

3.       Specific Objectives and Contents:  

Contents

Unit I: Groups                                   (25)                                      

1.1    Algebraic system

1.2    Operation, Cayley`s tables properties of binary operations.

1.3    Semigroups, monoids

1.4    Equivalence relation, quotient structures.

1.5    Definition and example of group

1.6    Integral power of elements of a group

1.7    Cyclic groups

1.8    Composition table

1.9    Elementary properties of groups and cyclic groups

1.10 Permutations and product of        permutations

1.11Group of permutations, symmetric group Sn, and dihedral group.

1.12Group actions.

Unit II: Subgroups                     (15)                       

2.1 Definitions and examples of subgroups.

2.2 Centralizer, normalizer

2.3 Properties of Subgroup5

2.4 Generators and defining relations.

2.5 Subgroups generated by subsets 5f a group

2.6 Lattice of subgroup.

Unit III: Normality, Co-sets, Quotient Groups and Homomorphism and Direct Products                                    (25)

1.1.         Co-sets, quotients of groups, normality and homomorphism.

1.2.        Algebra of subsets of group co-sets.

1.3.        Properties of homomorphism.

1.4.        Normalizer, stabilizer, centralizer orbits, Lagrange`s theorem.

1.5.        Counting principle.

1.6.        Isomorphism theorem: fundamental theorem, diamond and quotient isomorphism theorems and correspondence theorem auto-morphism.

1.7.        Direct Products.

Unit IV: Rings, Subrings, Ideals and Homomorphisms    (30)               

4.1. Definition and examples of rings and subrings.

4.2 Ideals and homomorphism

4.3 Algebra of ideals

4.4 Homomorphism of rings

4.5 Embedding and extension of rings.

4.6 Prime, maximal, nil-point and nil ideals.

4.7 Factorization domain.

4.8 Euclidean domain.

4.9 Direct products and direct sum of rings and ideals.

4.10 Principle ideal domain.

4.11 Unique factorization domain.

4.12 Properties of factorization domain

4.13 Ring of factorization

Unit V: Polynomial Rings               (15)

5.1 Definition and examples of polynomials division algorithm

5.2 Factorization of polynomials

Unit VI: Sylow’s Theorems and Classi-fication of Finite Groups   (15)

6.1 Group actions on a set.

6.2 Conjugate relations.

6.3 Cauchy’s theorems

6.4 Sylow’s theorems

6.5 Classification of finite groups

Unit VII: Fields                                     (25)

7.1 Algebraic extent of fields

7.2 Irreducible polynomials and Eisenstein criteria

7.3 Adjunction of roots

7.4 Algebraic extension

7.5 Algebraic closed fields.

7.6 Normal and separable extensions

7.7 Splitting fields

7.8 Normal extensions

7.9 Multiple roots

7.10 Finite fields

7.11 Separable extensions

1.       Instructional Techniques

4.1 General Instructional Strategies

Because of the theoretical nature of the course, teacher-centered instructional techniques will be dominant in the teaching-learning process. The teacher will adopt the following techniques:

·         Lecture with discussion

·         Use of software (math lab, mathematical if possible)

·         Investigative approach in problem solving

4.2 Specific Instructional Strategies

Unit

Chapter

Instructional techniques

I

Groups

Expository, Discussion and presentation

II

Subgroups

Expository, Discussion and presentation

III

Normality, co-sets, Quotient Groups and Homomorphism and Direct Products

Expository, Discussion and presentation

IV

Rings, subrings, ideals and Homomorphisms

Project work, Presentation

V

Polynomials Rings

Expository, Discussion and presentation

VI

Sylow’s theorem and classification of finite groups

Project work, Home assignment

VII

Fields

Class work, project work and assignment

 

2.       Evaluation

Students will be evaluated on the basis of the written classroom test in between and at the end of the academic session, the classroom participation, presentation of the reports and other practical activities. The scores obtained will be used only for feedback purposes. The Office of the Controller of the Examinations will conduct the annual examination at the end of the academic session to evaluate the student` performance. The types, numbers and marks of the subjective and objective questions will be as follows:

Types of questions

Total questions to be asked

Number of questions to be answered and marks allocated

Total marks

Group A: Multiple choice items

20 questions

20 × 1 marks

20

Group B: Short answers questions

8 with 3 ‘or’ questions

8 × 7 marks

56

Group C: Long answers questions

2 with 1 ‘or’ questions

2 × 12 marks

24

 

3.       Recommended books and references

Recommended books

Bhattarai, B.N. (2011) Introduction of Group Theory, Kathmandu: Subhakamana Prakashan

Bhattarai, B.N. (2011) Introduction of Rings and Modules, Kathmandu: Subhakamana Prakashan

Dummit, D.S. & Foote R. (2002). Abstract algebra, New Delhi: Wiley India Reprint

Fraleigh, J.B. (2003). A first course in abstract algebra, India: Pearson Education Inc.

Herstine, I.N. (1986). Abstract algebra, New York: Macmillan Publishing Company

Koirala, S.P. & Bhattarai B.N. (2010) A textbook on higher algebra, Kathmandu: Pragya Prakashan

References

Durbin, J.R. (2005) Modern algebra, India: John Wiley and Sons Inc.

Hersteine, I.N. (2008) Topics in algebra, New Delhi: Wiley, India.

Maharjan, H.B. (2000) First course in abstract algebra. Kathmandu: Ratna Pustak Bhandar.

Maharjan, H.B. (2007) Group theory, Kathmandu: Bhundi Puran

Maharjan, H.B. (2008) Rings and modules, Kathmandu: Bhundi Puran

Shrestha, R.M. (2006) Elementary linear algebra, Kathmandu: Sukunda Pustak Bhawan

Stheth, I.H. (2002) Abstract algebra, New Delhi: Prentice Hall of India

Thomas, W.H. (1974) Algebra, New York: Springer Verlag Inc.