Faculty: Satyabrata Sadangi

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), 2^nd 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

Another Counting Principle
https://www.youtube.com/watch?v=P6vYubWJak0&t=68s

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

Sylow’s Theorems.
https://www.youtube.com/watch?v=KSDpmZUpYqU

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 9

Definition and Examples of Rings
https://www.youtube.com/watch?v=j_f7O-4Rb9U&list=PLi01XoE8jYoi3SgnnGorR_XOW3IcK-TP6&index=24

Session 10

Some special classes of ring
https://www.youtube.com/watch?v=_RTHvweHlhE&list=PLi01XoE8jYoi3SgnnGorR_XOW3IcK-TP6&index=25

Session 11

The pigeonhole Principle
https://www.youtube.com/watch?v=IeTLZPNIPJQ

Session 12

Homomorphisms
https://www.youtube.com/watch?v=XPF5fe1WdKY&list=PLi01XoE8jYoi3SgnnGorR_XOW3IcK-TP6&index=11

Session 13

Ideals in ring
https://www.youtube.com/watch?v=F0wA0xLZSQ8

Session 14

Quotient rings
https://www.youtube.com/watch?v=Vqrd2nqD9Qc

Session 15 & 16

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
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

Euclidean Rings
https://www.youtube.com/watch?v=bNQeqVYKa1Y

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

Polynomial Rings
https://www.youtube.com/watch?v=J_eV53rHqDY

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

Introduction to field
https://www.youtube.com/watch?v=KCSZ4QhOw0I

Session 33

Extension Fields
https://www.youtube.com/watch?v=9hmr_Fjot_8

Session 34

Roots of polynomials
https://www.youtube.com/watch?v=WRkgMDVIETE

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 38

Elementary Basic Concepts of Vector Space
https://www.youtube.com/watch?v=TgKwz5Ikpc8&list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab&index=15

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

Session 41 & 42

Project 6:

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

Case Studies

Case Studies