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# Code(Credit) : CUTM1538(3-0-1)

## Course Objectives

• A major objective is to introduce students to the language and precision of modern algebra. This means that the course will be proof-based, in the sense that students will be expected to understand, construct, and write proofs.
• A challenge for all students of mathematics is to balance the understanding with the communication. There is a tendency to think you are finished once you see why a mathematical statement is true or false.

## Course Outcomes

 COs Course outcomes Mapping Cos with POs (High-3, Medium-2, Low-1) CO1 ·       Effectively write abstract mathematical proofs in a clear and logical manner. PO1(3), PO2(2) CO2 ·       Locate and use theorems to solve problems in number theory and theory of polynomials over a field. PO2(2), PO4(2) CO3 ·       Demonstrate ability to think critically by interpreting theorems and relating results to problems in other mathematical disciplines. ·       Demonstrate ability to think critically by recognizing patterns and principles of algebra and relating them to the number system. PO9(3) CO5 ·       Work effectively with others to discuss homework problems put on the board. PO2(2), PO1(3)

## Course Syllabus

MODULE – I (6hr+0hr+2hr)

Group Theory:

Another Counting Principle, Sylow’s Theorems.

Project 1 : A Notes on the Proof of the Sylow Theorem

MODULE – II (6hr+0hr+2hr)

Ring Theory:

Introduction to Ring, Some special classes of ring, Ring homomorphisms.

Project 2: A study on ring theory and it’s property

MODULE – III (3hr+0hr+2hr)

More Ideals and Quotient Rings, The Field of Quotients of an Integral Domain.

Project 3: The Quotient Field of an Intersection of Integral Domains

MODULE – IV (4hr+0hr+2hr)

Euclidean Rings, A Particular Euclidean Ring, Polynomial Rings.

Project 4: On the Existence of a Euclidean Algorithm in Number Rings with Infinitely Many Units

MODULE – V (4hr+0hr+0hr)

Polynomial Rings over the Rational Field, Polynomial Rings over Commutative Rings.

MODULE – VI (3hr+0hr+2hr)

Fields:

Extension Fields, Roots of polynomials

Project 5: A study on Structure of a Finite Field

MODULE – VII (4hr+0hr+2hr)

Vector Spaces:

Elementary Basic Concepts of Vector Space, Linear Independence and Basis, Dual Spaces, Inner Product Spaces

Project 6: Notes on dual spaces

BOOK PRESCRIBED

Topics in Algebra – I. N. Herstein (John Wiley and Sons or Vikas Publication), 2nd Edition

Chapters: 2 (2.11 to 2.12), 3 (3.1 to 3.11), 4 (4.1 to 4.4), 5(5.1 and 5.3)

BOOKS FOR REFERENCE

1. Singh and Q. Zameeruddin, Modern Algebra, Vikas Publishing House, 1990
2. B. Bhattacharya, S. K. Jain and S. R. Nagpal, Basic Abstract Algebra, Cambridge University Press, 1995.

## Session 1

Introduction to Group Theory

## Session 3

Theorems on Another Counting Principle

## Session 4

Application of theorems on Another Counting Principle (Lemma)

https://www.slideshare.net/Vaishnavi_0211/group-theory-57404640

## Session 6

Theorems and it’s proof of Sylow’s Theorems

## Project 1:

A Notes on the Proof of the Sylow Theorems

https://www.math.drexel.edu/~rboyer/courses/math533_03/sylow_thm.pdf

## Project 2:

A study on ring theory and it’s property’s

http://ignited.in/I/a/89154

## Session 17

More Ideals and Quotient Rings

## Session 18

The Field of Quotients of an Integral Domain

## Session 19

Theorems on the Field of Quotients of an Integral Domain

## Project 3:

The Quotient Field of an Intersection of Integral Domains

## Session 23

Theorems on Euclidean Rings

## Session 24

A Particular Euclidean Ring

## Project 4:

On the Existence of a Euclidean Algorithm in Number Rings with Infinitely Many Units

https://www.math.ucdavis.edu/files/2515/5197/6414/Jake_Parkhurst_Senior_Thesis.pdf

## Session 28

Polynomial Rings over the Rational Field

## Session 29

Theorems on Polynomial Rings over the Rational Field

## Session 30

Polynomial Rings over Commutative Rings

## Session 31

Theorems on Polynomial Rings over Commutative Rings

## Project 5:

A study on Structure of a Finite Field

http://abstract.pugetsound.edu/aata/section-finite-field.html

## Session 37

Introduction to vector space

## Session 39

Linear Independence and Basis

## Session 40

Dual Spaces and Inner Product Spaces

## Project 6:

Notes on dual spaces

https://sites.math.northwestern.edu/~scanez/courses/334/notes/dual-spaces.pdf