Welcome to the course page for SMA205 - Introduction to algebra
In case you have any questions, please feel free to email me (see page 1 of the lecture notes for my email address).
News concerning the course
The course will now be run by Mr James Nkuubi, because I have to go back to England for my PhD.
It seems like you have done very well on the CAT and on the first Home Assignment. You will get them back after the Christmas break.
All lecture notes are now (finally) posted, see below. Make sure you have all pages (page 1 to 77) when preparing for the exam.
The timetable is as follows:
Mondays, 11am-12noon. Location: MH1
Wednesdays, 8am-10am. Location: AGRI LAB
The examination will be as follows:
- Continuous assessment test (10% of the total grade).
- Home assignment 1. (10% of the total grade). On Induction, Congruences and Factorization.
- Home assignment 2. (10% of the total grade). On RSA cryptography. This is distributed in class on Thursday, Dec 14. Deadline: January 29.
- Final exam (70% of the total grade). It is now confirmed that the final exam will take place in the regular exam period with all your other exams. The final exam will be much harder than the CAT, you are expected to:
- understand all the proofs in the posted lecture notes
- be able to solve nontrivial problems (like the problems in the lecture notes)
Course contents and lecture notes
The course consists of five parts. Lecture notes will be posted here and distributed to the class representatives in the lectures.
To get the full set of notes in one document, click here. However, this could take quite some time to download.
- Part 1: Review of sets, relations and functions.
- Part 2: Elementary number theory
- Part 3: Proofs and problem-solving Lecture notes (page 51 to 58)
- Part 4: Applications to cryptography Lecture notes (page 59 to 65)
- Information on the exam. Information sheet (page 66)
- Part 5: Introduction to abstract algebra Lecture notes (page 67 to 77)
All you really need for this course is the lecture notes above. However, for those of you who are interested in more reading, I have collected some references to material on the Internet and in the Chiromo Library.
There are many online references for the material in this course, some very good, some not so good. Here is a selection of the best notes I am aware of. Note that notation may vary between different texts.
General references for the whole course:
If you need a particular definition, you can probably find it by searching on Wikipedia.
An excellent little book, available for free online, is Paul Garrett's Introduction to abstract algebra. This book covers most of what we will do in this course. If you want an alternative to the lecture notes, you should definitely consult this text.
Sets, relations and functions are covered in section 3 of Garrett's book, already mentioned above.
Some general notes on sets can be found here.
If you want to compute the GCD or LCM of two integers, you can do it here. (You just have to enter 2 in the field "Total numbers" before you can enter the two integers.)
A similar page, which gives you the prime factorization of an integer, is here. When you use these pages, remember that you must also be able to do these kinds of calculations by hand on the exam.
There is an excellent e-book by Robert D. Carmichael covering more or less the same number theory as we do in this course. This book has been made available online for free through the Gutenberg project.
William Chen has written a very good set of notes on Elementary Number theory. These notes cover much more than we will have time for in this course.
You can also have a look at some notes by Bruce Ikenaga. These are in Postscript format.
Another book is Elementary number theory by W. Edwin Clark, also in Postscript format.
Here is a link to a survey of elementary number theory with special emphasis on how to solve hard problems. Written by Naoki Sato.
To read more about various methods of proof, you can check out these notes.
If you want to learn more about cryptography, the RSA labs has compiled a superb introduction in the form of a FAQ page. You can also download the whole FAQ as a pdf file (1.4 MB). In these pages, you also find a great number of references to further reading.
Here is a list of the RSA labs Challenge numbers which give you from US$ 30,000 if you can find their prime factorization. There is also an associated FAQ.
The well-known number theorist William Stein has written some very good lecture notes on number theory which among other things cover material on factorization and cryptography, more advanced than in this course. These notes also give a very good introduction to Elliptic curves, which is an important and very active number-theoretic research area, with applications in cryptography.
A good reference is the book by Edwin H. Connell
There is also a introductory book by W. Edwin Clark in Postscript format.
For the student who wants to learn more about abstract algebra, I would recommend the book Topics in Algebra, by Herstein. This is a superb introduction to abstract algebra, more advanced than this course, but still on an accessible level. Reading this book after the present course would be a very good preparation for future algebra courses. There are at least four copies in the library.
Here are some other books that can be found in the library, for the interested students. Most of them are more advanced than this course. They are ordered roughly in increasing order of difficulty, within each category.
Carmichael: The theory of numbers (same book as the e-book mentioned above).
Bolker: Elementary number theory: an algebraic approach.
Tattersall: Elementary Number Theory in Nine Chapters.
Rose: A course in number theory.
Davenport: The higher arithmetic.
Fraleigh: A First Course in Abstract Algebra.
Jordan and Jordan: Groups.
van der Waerden: Modern Algebra, volume I and volume II.
Bahturin: Basic structures of Modern Algebra.
Reid: Undergraduate commutative algebra.
Other related topics:
Reid: Undergraduate algebraic geometry.
Goodaire and Parmenter: Discrete mathematics.