Advanced Algebra

Teacher

Satyabrata Sadangi

Category

Core Courses

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Course Name : Advanced Algebra

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.

Learning Outcomes

Effectively write abstract mathematical proofs in a clear and logical manner.

Locate and use theorems to solve problems in number theory and theory of polynomials over a field.

Demonstrate ability to think critically by interpreting theorems and relating results to problems in other mathematical disciplines.

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 Plan

Session 1

Introduction to Group Theory

https://www.youtube.com/watch?v=zkADn-9wEgc

Session 2

Session 3

Theorems on Another Counting Principle

https://www.youtube.com/watch?v=qJ7AYDmHVRE

Session 4

Application of theorems on Another Counting Principle (Lemma)

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

Session 5

Session 6

Theorems and it’s proof of Sylow’s Theorems

https://www.youtube.com/watch?v=tVxjM1KuXBg

Session 7 & 8

Project 1:

A Notes on the Proof of the Sylow Theorems

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

Session 11

Session 15 & 16

Project 2:

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

http://math.bu.edu/people/svh/RingTheoryMathcamp.pdf

Session 17

More Ideals and Quotient Rings

https://www.youtube.com/watch?v=Y6P6-Wp5tiI

Session 18

The Field of Quotients of an Integral Domain

https://www.youtube.com/watch?v=4F2QB2LGR9o

Session 19

Theorems on the Field of Quotients of an Integral Domain

https://www.youtube.com/watch?v=tBTC2f081SI

Session 20 & 21

Project 3:

The Quotient Field of an Intersection of Integral Domains

https://core.ac.uk/download/pdf/82478993.pdf

Session 22

Session 23

Theorems on Euclidean Rings

https://www.youtube.com/watch?v=myCq2IOBvB4

Session 24

A Particular Euclidean Ring

https://www.youtube.com/watch?v=b991l-C88iM

Session 25

Session 26 & 27

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

https://www.youtube.com/watch?v=Es37gGo5F4Y

Session 29

Theorems on Polynomial Rings over the Rational Field

https://www.youtube.com/watch?v=gfAIyUfcSQ0

Session 30

Polynomial Rings over Commutative Rings

https://www.youtube.com/watch?v=ahUEy2FGulo

Session 31

Theorems on Polynomial Rings over Commutative Rings

https://www.youtube.com/watch?v=rhw2hoasa6Q

Session 32

Session 33

Session 34

Session 35 & 36

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

https://www.youtube.com/watch?v=ozwodzD5bJM

Session 39

Linear Independence and Basis

https://www.youtube.com/watch?v=OLqc_rt7abI

Session 40

Dual Spaces and Inner Product Spaces

https://www.youtube.com/watch?v=SjCs_HyYtSo

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Case Studies

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    Mr. Satyabrata Sadangi has received his M.Sc. (Mathematics) from National Institute of Technology(NIT) Rourkela, Orissa, India and B.Ed. from PRSU University, C.G, India. Mr. Sadangi has more than ten years of teaching experience in different educational institutes. Presently he is working as an Assistant Professor, Department of Mathematics, School of Applied Sciences, Centurion University of […]