# News

14.08: First lecture Tuesday 22 of August, at 14:15 in K26 (are located in Kjemiblokk 4, 1.etg).

# General information

Lecturer: Sverre Smalø, email: sverresm at math ntnu no, SBII, room 850, tlf. 735 91750

Office hours: Tuesday 13:15. I am awailable all days between 9.00 to 15.00 from August 22nd on and until December ?th. (Only by appointment for Saturdays and Sundays)

Teaching hours:

 Lectures: Tuesday 14:15-16:00, K26 Friday 14:15-16:00, R60

Some of the Friday-lectures will be used as exercise sessions.

# Textbook and syllabus

Textbook

 Authors: P. B. Bhattacharya, S. K. Jain and S. R. Nagpaul Title: Basic Abstract Algebra Publisher: Cambridge University Press Edition: Second Edition ISBN: 0-521-46629-6

Preliminary syllabus:

 Chapter 9 All sections Chapter 10 All sections Chapter 14 14.1-14.5 Chapter 19 19.1-19.3 Chapter 20 All sections Chapter 21 All sections

# Plan/log

 Date Themes Reference 22.08 Introduction, definitions, various examples including ring of polynomials, rings of formal power series, rational functions and Laurent series. Chap 9.1, 9.2, 9.3 25.08 Subrings, characteristic of a ring, idempotent og nilpotent elements, ideals and homomorphisms Chap 9.4, 9.5, 10.1, 10.2 29.08 More on Ideals and homomorphisms, left ideals, right ideals, ideals in the full matrix algebra, Intersections and sums af ideals, Chap 10.1, 10.2, 10.3 01.09 Exercise 1 in the first hour! Direct sum of Ideals, Minimal, maximal and prime ideals, the real numbers as the rings of Cauchy-sequences modulo the ideal of sequences converging to zero Chap 10.3, 10.4 05.09 The Chinese remainder theorem, Maximal ideals are prime, the maximal ideals of Z, C[x] and R[X], description of prime ideals for commutative rings. Chap 10.5,10.6 08.09 Nilpotent and nil ideals, Zorns's lemma and the existence of maximal ideals. Modules and Vectorspaces Chap 14.1, 14.2 12.09 The rest of Chapter 14.2, and most of 14.3 Chap 14.2, 14.3 15.09 Exercises 2 and 3, the rest of 14.3 and start of 14.4 Chap 14.3, 14.4 19.09 Rest of 14.4 and started on 14.5 Chap 14.4, 14.5 22.09 Finished 14.4 and 19.1, Started on 19.2 Chap 14.5, 19.1 19.2 26.09 Chap 19.2 29.09 We will take a look at the exercises, so look through the problems in exercise 4. Any question from exercise 1, 2 and 3 can also be raised. 03.10 Endomorphism ring of a direct sum as a matrix ring, Th 19.2.7, 19.2.8,19.2.9, 19.2.10 Chap 19.1, 19.2 06.10 Modules of finte length and composition series.th 19.2.11 10.10 Wedderburn-Artin + examples Chap 19.3 13.10 Exercise 5 and 6 17.10 More examples illustarting Wedderburn-Artin's theorem and modules of finite length. Prove that for semisimple modules artinian and noetherian is the same 20.10 Exercise 6 24.10 Smith normal form Chapter 20.1,2,3 27.10 Exercise 6, 7, 8 31.10 Applications of Smith normal form Chapter 21.1,2,3 03.11 Exercise 8, 9 07.11 Finishing Rational canonical forms and Jordan canonical forms Chapter 21.4,5 10.11 Do the exam from Dec. 2016 14.11 Summary

# Exam

Final exam: 18.12.2016, written, 4 hours, 9:00-13:00

# Problem sessions(tentative)

 Exercise 1 Chapter 9, page 173-176: 1, 2, 3, 4, 5, 7, 9, 11 Chapter 10, page 187: 1, 2, 4, 7 Exercise 2 Chapter 10, page 194-195: 1, 2, 3, 4, 8, 10, 11, 12 page 202-203: 1, 2, 3, 4, 7 Exercise 3 Chapter 10, page 209: 1, 2, 4, 6 page 210: 1, 2 Exercise 4 Chapter 14, page 252-253: 1, 3, 4, 5, 8, 10, 11 page 260: 2, 3, 4, 6, 7 Exercise 5 Chapter 14, page 262-263: 1, 2, 3, 5, 6 page 268: 1, 2, 4, 5, 7, 8, 9 Exercise 6 Chapter 19,1, page 368: 1, 2 19,2, page 381: 1, 4, 6 Exercise 7 Chapter 19,2, page 381-382 2, 7, 8, 9, 11 Exercise 8 Chapter 19,3, page 388 3, 5 chapter 20,3, page 401 1,2,3 Exercise 9 Exam Dec 11 2007 Problems 1, 2, 3 Nov 30 2005 Problems 1, 4 Exercise 10 Dec 11 2007 Problem 4 Nov 30 2005 Problem 2, 3 Dec 11 2009 Problem 2, 3 Exercise 11 Dec 13 2010 Dec 2011 dec 2012 dec 2014 dec 2014