UM 203. Introduction to Algebraic Structures - Spring 2019 - Vamsi Pingali

Unfortunately, no single book will do. The texts we will be following are as follows. The first two will be our main textbooks.

Instructor : Vamsi Pritham Pingali,

Office : N23 in the mathematics building. (It might be a good idea to email me if you plan on coming to my office.)

Teaching assistants : K. Hariram ( and Abhash Kumar Jha (

Classroom and timings : Tuesdays and Thursdays 5:30-6:30 PM and Wednesdays 12-1 in Room G2 (Old physics building).

The Grading policy : 15% for Tests based on HW, Midterm-35%, and 50% for the Final. Under NO circumstances will makeup exams be held for the midterms. If you have a valid and provable excuse, (Schedule conflicts with other courses do NOT constitute as valid excuses. You are supposed to resolve them before registering for the courses.) then your performance on the other exams shall determine your grade on your midterms.

Exams : The Midterm will be held on 18 Feb from 2-4 pm. The final examination will be held on 22 April (Monday) from 2-5 pm. The syllabus for the final is everything covered in the course.

Ethics: Read the information on the IISc student ethics page. In short, cheating is a silly thing. Don't do it. As for homeworks, write them up on your own. You are allowed to discuss them amongst yourselves but please write the solutions on your own. That said I must hasten to add that you learn mathematics best when you solve the problems entirely by yourself.

Here is the tentative schedule. (It is subject to changes and hence visiting this webpage regularly is one of the best ideas in the history of best ideas.)



 Syllabus to be covered


1 Jan - 7 Jan

Logistics, Naive set theory done right (Tuesday notes); Axiom of Choice, Equivalence relations, Involutions (Wednesday notes); Partial orders and Zorn's lemma, Cardinality (Thursday notes)


8 Jan - 14 Jan

Integers and Rationals (Tuesday notes), Complete induction and the pigeon-hole principle(Wednesday notes), Permutations and Combinations (Thursday notes)


15 Jan - 21 Jan

Cycles in permutations (Tuesday notes), Ordinary generating functions (Wednesday notes), Combinatorics and O.G.Fs (Thursday notes)


22 Jan - 28 Jan

Exponential generating functions, Graphs and the handshaking lemma (Tuesday notes), Trails, walks, Adjacency matrices (Wednesday notes), Eulerian tours (Thursday notes)


29 Jan - 4 Feb

Symmetric differences, Trees -> Covered by Siddhartha Gadgil (Tuesday notes (a rough sketch of my take on Siddhartha's material)), Planar graphs (Wednesday notes ), Fundamental theorem of arithmetic and Euclid's algorithm (Thursday notes )


5 Feb - 11 Feb

Linear Diophantine equations, Least common multiple, Infinitude of primes (Tuesday notes), Modular arithmetic (Wednesday notes), Definitions and examples of Rings and Fields (Thursday notes)


12 Feb - 18 Feb

Units and zero divisors in Z/mZ, definition of Integral Domains (Tuesday notes), Fermat's and Euler's theorems (Wednesday notes), Wilson's theorem, Pythagorean primes using the Gaussian integer ring(Thursday notes) ,Midterm on 18 Feb at 2:00 pm (The syllabus is everything up to and including week 6, i.e., up to and including the definitions and examples of rings and fields.)


19 Feb - 25 Feb

Midterm week


26 Feb - 4 March

Frobenius property, a version of Euler theorem for squarefree integers, definition of Ring homomorphisms (Tuesday notes), Examples of ring homomorphisms and characteristic of a commutative ring (Wednesday notes), Frobenius endomorphism (Thursday notes)


5 March - 11 March

Diffey-Hellman protocol, RSA, Chinese Remainder theorem (Tuesday notes), Products of rings and the Chinese Remainder theorem, Polynomial rings (Wednesday notes), Division theorem and Euclid's algorithm for polynomials (Thursday notes)


12 March - 18 March

Fundamental theorem of arithmetic for polynomials, congruences, and Diophantine-like equations for polynomials (Tuesday notes), Chinese Remainder theorem for polynomials (Wednesday notes), Fundamental theorem of symmetric polynomials and cubic equations (Thursday notes)


19 March - 25 March

Solving the quartic and the Abel-Ruffini theorem for quintics (Tuesday notes), The Dihedral group and Cyclic groups (Wednesday notes), Subgroups of cyclic groups, Cayley's theorem (Thursday notes)


26 March - 1 April

Generating cycles in Sn, Lagrange's theorem for groups (Tuesday notes), Normal subgroups and quotient groups (Wednesday notes), Cauchy's theorem and the first isomorphism theorem(Thursday notes)


2 April - 8 April

Quadratic residues - Reduction to prime powers moduli (Tuesday notes), Quadratic residues - Reduction to prime moduli and definition of the Legendre symbol (Wednesday notes), Quadratic residues - Rules for manipulating the Legendre symbol and Euler's criterion (Thursday notes)


8 April - 15 April

Quadratic residues - Quadratic Reciprocity (Done on Tuesday from Childs' book). No more classes.

HW - Kyon ki teacher bhi kabhi student tha....


 Test to be held on

 Homework (subject to changes; please check regularly)


11 Jan

HW 1


25 Jan

HW 2


8 Feb

HW 3


15 Feb (Note this !)

HW 4


8 Mar

HW 5


15 Mar (Note this!)

HW 6


29 Mar

HW 7


5 Apr (Note this)

HW 8