MVE150/MMG500 Algebra

MVE150/MMG500 Algebra

This page contains the program of the course: lectures and exercise sessions.

Lecturer and examiner: Per Salberger.


The schedule of the course is in TimeEdit.


Normally there will be a lecture Tuesday 8-10 and Thursday 13-15. The first Tuesday January 21 st there is also a lecture 10-12.


J.R. Durbin: Modern Algebra, An Introduction, John Wiley & Sons, version 5 or 6.

Note that the sections 53, 54, 55 in the fifth edition are moved to sections 56, 57, 58 in the sixth edition!

Problem sessions The problem sessions are held on Tuesday 10-12 and Friday 10-12, beginning

January 24 th . The problems will be chosen from the list of recommended exercises on the home

page. All these are important, and most of them will be discussed during the problem sessions. For

the English spoken problem sessions by Jiacheng Xia, the classroom is EL 43 on Tuesdays and EL 42

on Fridays. For the Swedish spoken sessions by Per Salberger, it is in MVH 12 for both.

Recommended exercises

The exercises below are selected from the sixth edition of Durbin’s book . The same exercises may also

be found in the fifth edition and usually with the same numerotation. The exceptions are listed with a V

for their numbers in the fifth edition. A star * indicates that the exercise is considered more

challenging. Please try to do as many exercises as possible before the days indicated for them!


Days Exercises
Jan 24 3.1-3.8, 3.13, 3.24, 4.1, 4.2, 5.1-5.12, 5.17, 6.1-6.3, 6.5, 6.9.
Jan 28 & 31 7.1, 7.2, 7.13, 7.14, 7.22*-7.24*, 9.5-9.8 , 10.1-10.3 , 10.13 , 10.25, 10.26, 11.3, 11.13-11.17*, 12.7 , 12.20 , 12.21.
Feb 4 & 7 13.6, 13.7, 13.9, 13.19*, 13.20, 13.21, 14.3-14.7, 14.9, 14.18, 14.25, 14.14, 14.23, 14.24, 14.31, 14.32, 14.33, 14.34, 15.24*, 16.2, 16.5, 16.7, 16.15, 17.2, 17.4, 17.7, 17.10, 17.12, 17.24, 17.25, 17.27, 17.30, 17.32.
Feb 11 & 14 15.7, 15.9, 15.12, 15.17-15.19, 15.21, 17.14, 17.15*, 18.11*-18.14, 19.1, 19.2-19.5, 19.7, 19.14, 19.15, 19.21=V19.19, 19.22=V19.20, 19.23=V19.21, 19.27=V19.25, 19.32=V19.30, 19.34=V19.32, 21.5-21.10, 21.15-21.18, 21.26-21.28*, 21.34, 21.35.
Feb 18 & 21 22.1, 22.2, 22.6, 22.12, 22.14, determine the quotient groups: (a) 4Z/12Z (b) R # /R >0 , 23.2, 23.4, 23,12, 23.12, 23.13, V23.20*, 56.5= V53.5, 56.6= V53.6, 56.8= V53.8, 56.9= V53.9, 57.5= V54.5, 57.13= V54.13, 58.1= V55.1, 58.2=V55.2, 58.7= V55.7.
Feb 25 & 28 25.1, 25.4, 25.5, 25.7, 25.21, 26.1-26.9, 26.13, 26.18, 26.23*, 27.20*, 27.21, 27.23, 34.6, 34.8, 35.4-35.6, 35.9, 35.12, 35.13, 35.18, 36.1, 36.9, 36.11, 36.12, 36.22*-36.24*, 37.2-37.5, 37.8.
March 3 & 6

38.1-38.4, 38.9, 38.12-38.15, 38.18*-38.20*, 38.22, 38.24*, show that any R-ideal I which contains an invertertible element is equal to R, 39.1-39.4, 39.7, 39.8*-39.10*, 39.12*, 41.2,41.3, 41.12, 42.4, show that any field of characteristic 0 contains a subfield isomorphic to Q, determine the minimal polynomials for √2+√3 and i=√-1 over Q, 40.6-40.10, 50.3=V45.3, 50.4=V45.4, 50.5=V45.5.

March 10 43.1= V44.1, 43.18= V44.18, 30.3, 30.8, 35.14. Let a≠0 and f(x) be a polynomial such that f(xn) is divisible by (x-a). Show that f(xn) is also even divisible by (xn -an).

Solutions to some exercises: Solution to the exercise on Cube.docx


Solution to the Home Exam (March 21st)

Solutions Algebra home exam March 2020.pdf

New (March 12th)!

As the written exams will be carried out at home there will no longer
be exercises where the students should reproduce proofs from Durbin's 
book. The two exercises 5 and 6  will instead be problems of the same
kind as exercise 1- 4.
The president of Chalmers has, as is well known, decided that all hall exams during the exam week 
will be done at home. Here are some guidelines.
Rules for the exam (March 13th)
The examination must be conducted individually, that is, cooperation is not allowed. Due to the changed circumstances, all aids are allowed regardless of what is written on the exam. Checks on plagiarism will be carried out.
You have 4 hours to complete the exam. Solutions are written on paper, or digitally on a digital writing pad if you have access to it. Never write more than one task on each sheet. After 4 hours, you have 30 minutes to scan / photograph your solutions and organize and submit your solutions according to one of the following ways in order of priority:

1. A single pdf file where the pages are arranged in the order of the questions

2. Image files (jpg or png) or pdf files where each file contains solution to just one question and named according to "Question 1", or "Question 1 page 1", Question 1 page 2 "etc if there are multiple pages to a question. You who have a certificate for extended time will have 6 hours to complete the exam and after that 30 minutes for you to submit according to the same instructions as above. When you submit, it will be stated that the exam is submitted late and if you have not already notified the examiner that you are entitled to extended time, you should do so afterwards.

NEW INFORMATION! Information about the Algebra exam 2020.docx

There is a written exam in March. There are two re-exams in 2020. There are four exercises in the

exam about applying basic theory in the course, and two exercises about stating important

definitions and theorems and giving proofs to variants of them. Each exercise is worth 4 or 5 points

and there are 25 points in total. No extra aid (textbook or internet) is allowed during the exam.


For examples of previous exams, click here. MMG500 Mar 2017.pdf, MMG500 Jun 2017.pdf

MMG500 Aug 2017.pdf, MMG500 Mar 2018.pdf, MMG500 Jun 2018.pdf

MMG500 Aug 2018.pdf, MMG500 Mar 2019.pdf,  Exam MMG500 Jun 2019.pdf,

Exam MMG500 Aug 2019.pdf

For solutions of exams in 2019, click here. 

Solutions MMG500 Mar 2019.pdf,  Solutions MMG500 June 2019.pdf, 

Solutions MMG500 Aug 2019 corrected.pdf (New Version!)


The dates, times and places for the exams are in the student portal (Links to an external site.).

Examination procedures

In Chalmers Student Portal (Links to an external site.) you can read about when exams are given

and what rules apply to exams at Chalmers. In addition to that, there is a schedule (Links to an

external site.) when exams are given for courses at the University of Gothenburg.

Before the exam, it is important that you sign up for the examination. You sign up

through Ladok (Links to an external site.)

At the exam, you should be able to show valid identification.

After the exam has been graded, you can see your results in Ladok.

At the annual (regular) examination: 

When it is practical, a separate review is arranged. The date of the review will be announced here on

the course homepage. Anyone who can not participate in the review may thereafter retrieve and

review their exam at the Mathematical Sciences Student office (Links to an external site.). Check

that you have the right grades and score. Any complaints about the marking must be submitted in

writing at the office, where there is a form to fill out.

At re-examination: 
Exams are reviewed and retrieved at the Mathematical Sciences Student office (Links to an external

site.). Check that you have the right grades and score. Any complaints about the marking must be

submitted in writing at the office, where there is a form to fill out.

Course evaluation

At the beginning of the course, at least two student representatives should have been appointed to

carry out the course evaluation together with the teachers. The evaluation takes place through

conversations between teachers and student representatives during the course and at a meeting

after the end of the course when the survey result is discussed and a report is written. 

Guidelines for Course evaluation (Links to an external site.) in Chalmers student portal.

Student representatives

The following students have been appointed:

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Course summary:

Date Details Due