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.
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
- Singh and Q. Zameeruddin, Modern Algebra, Vikas Publishing House, 1990
- B. Bhattacharya, S. K. Jain and S. R. Nagpal, Basic Abstract Algebra, Cambridge University Press, 1995.
Session 4
Application of theorems on Another Counting Principle (Lemma)
https://www.slideshare.net/Vaishnavi_0211/group-theory-57404640
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 12
Session 19
Theorems on the Field of Quotients of an Integral Domain
Session 20 & 21
Project 3:
The Quotient Field of an Intersection of Integral Domains
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 29
Theorems on Polynomial Rings over the Rational Field
Session 31
Theorems on Polynomial Rings over Commutative Rings
Session 35 & 36
Project 5:
A study on Structure of a Finite Field
http://abstract.pugetsound.edu/aata/section-finite-field.html
Session 38
Elementary Basic Concepts of Vector Space
https://www.youtube.com/watch?v=TgKwz5Ikpc8&list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab&index=15
Session 41 & 42
Project 6:
Notes on dual spaces
https://sites.math.northwestern.edu/~scanez/courses/334/notes/dual-spaces.pdf
Case Studies
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Our Main Teachers
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 […]