
This semester I am teaching MA 812: AlgebraII.
This is a course for first year Ph.D. students, however, all Ph.D. students are encouraged to attend the classes.
Last semester, I had taught MA 811: AlgebraI and this course takes up from where we left in the last semester.
Algebra provides axiomatic structures; groups, rings, vector spaces, modules, to quote a few; that aid in studying other branches of mathematics.
Our aim in the course is to learn linear algebra, including symmetric and alternating forms, and we will conclude the course with representations of finite groups.
While studying linear algebra, we try to do it in general as much as possible. That is, we work over a general commutative ring for most of the time instead of the conventional setting of working over a field.
General information:
 It is necessary to have 80% of attendance in this course.
 The moodle page for the course should get ready soon.
Make sure you enroll for the course to receive all updates.
 You are all encouraged to meet me personally to discuss any difficulty that you may be facing.
You may drop me an email to take an appointment or just show up in my office.

Schedule:
All the lectures will be held from 11 am till 12.30 pm on Wednesdays and Fridays.
The venue is the room 105 in the mathematics department.

Textbook & reference books:
We are going to follow the Algebra book for the course.
Apparently there are multiple uses of it but we will concentrate on the first one.
There are many good books on the subject.
There are the classics by Jacobson (Basic algebra III and Lectures in abstract algebra IIII).
And then there are somewhat modern ones by Cohn (Basic alegbra and Further algebra and applications) and Knapp (Basic algebra and Advanced algebra).
You may also want to take a look at the encyclopaedia by Shafarevich on the subject.
You may follow any of these or any other book for problem solving and/or for reading more on algebra.

Courseplan:
This is the prescribed syllabus of the course.
We will try to complete this and will also try to learn something more than this.
Valuations and completions: definitions, polynomials in complete fields (Hensel's Lemma, Krasner's Lemma), finite dimensional extensions of complete fields, local fields, discrete valuations rings.
Transcendental extensions: transcendence bases, separating transcendence bases, Luroth's theorem. Derivations.
Artinian and Noetherian modules, KrullSchmidt theorem, completely reducible modules, projective modules, WedderburnArtin Theorem for simple rings.
Representations of finite groups: complete reducibility, characters, orthogonal relations, induced modules, Frobenius reciprocity, representations of the symmetric group.


Course calendar:
This course calendar is subject to revision during the semester.
Week 
Monday 
Tuesday 
Wednesday 
Thursday 
Friday 
Saturday 
Sunday 
1 
Jan 4

Jan 5
 Jan 6
§§ XII.1XII.3 
Jan 7

Jan 8
Adams.I 
Jan 9

Jan 10

2 
Jan 11

Jan 12

Jan 13

Jan 14
Tutorial 1 
Jan 15

Jan 16

Jan 17

3 
Jan 18

Jan 19

Jan 20
Tutorial 1 is due 
Jan 21

Jan 22

Jan 23

Jan 24

4 
Jan 25

Jan 26

Jan 27

Jan 28

Jan 29

Jan 30
Quiz 1 
Jan 31

5 
Feb 1

Feb 2

Feb 3

Feb 4

Feb 5

Feb 6

Feb 7

6 
Feb 8

Feb 9

Feb 10

Feb 11

Feb 12

Feb 13

Feb 14

7 
Feb 15

Feb 16

Feb 17

Feb 18

Feb 19

Feb 20

Feb 21

8 
Feb 2228
Midsemester examination 
9 
Feb 29

Mar 1

Mar 2

Mar 3

Mar 4

Mar 5

Mar 6

10 
Mar 7

Mar 8

Mar 9

Mar 10

Mar 11

Mar 12

Mar 13

11 
Mar 14

Mar 15

Mar 16

Mar 17

Mar 18

Mar 19

Mar 20

12 
Mar 21

Mar 22

Mar 23

Mar 24

Mar 25
Good Friday 
Mar 26
Quiz 2 
Mar 27

13 
Mar 28

Mar 29

Mar 30

Mar 31

Apr 1

Apr 2

Apr 3

14 
Apr 4

Apr 5

Apr 6

Apr 7

Apr 8
Gudhi Padwa 
Apr 9

Apr 10

15 
Apr 11

Apr 12

Apr 13

Apr 14

Apr 15

Apr 16

Apr 17

Tutorials:
I hear and I forget.
I see and I remember.
(Confucius)

We should aim to attempt each and every exercise of the book.
I will mark a few of them for weekly tutorials.
Five of those will need to be submitted by students for grading within the given time.
Examinations:
We will have two quizzes in addition to the usual midsemester and the endsemester examination.

Grading:
The distribution of 100 marks for this course will be as follows:
Seminars (5 + 5)
Quizzes (10 + 10)
Assignments (20)
Midsem (20)
Endsem (30)

Links:
 Companion to Lang by George Bergman.
(I have not read this carefully and so cannot comment on this, but I hope that this will be useful to you.
If it is not so, please let me know.)

