# Rings and number theory [Ringe og Talteori, MM510]

## Reexam

The date for the reexam is 23.6.2011.

## Exam solution sketches

I prepared some exam solution sketches for you. Let me add that 50 points are 100 percent.

## Exam announcement

The exam will take place on Thursday, 6.1.2011, from 9.00 to 13.00 in U81 and U82. There will be 10 questions with a total number of 85 points, and you will need 30 points to pass. So you have quite a choice, to show what you can do rather than what you cannot do. There will be questions according to the syllabus, where Sections 9.3 to 9.5 and of Sections 10.2 to 10.6 are excluded. The exam questions will be in english (so please bring a dictionary if necessary) - however, you may answer in either english or danish. On the MM510-2009-pages of Martin Svensson, there is some material you might want to look at, like old exam questions etc. All home assignments I have marked were satisfactory, their authors may take part.

## Slides / Course Log / Time Table

Click here for the most recent version [13.12.2010] of the set of my slides. At the moment, they are more suitable for viewing on the screen rather than printing, as you will get, for example, ten pages for just one ``incremental'' proof.

Here is a protocol of the course's actual progression, including exercise sheets. Plus a plan for the forthcoming lessons.

WeekEvents Content
45 (8.11.)* Mon 8-10 U28 & Wed 10-12 U37 Chapter 1. Induction principles. Further proof elements such as subproofs, proof by contradiction, uniqueness proofs; there are different proofs for the same statement. Detailled blackboard proofs of Theorem 1.1, Corollary 1.2, Theorem 1.3, Theorem 1.8, Theorem 1.11. Continued fractions, visualization of the Euclidean Algorithm by partitioning rectangles into squares (see slides). Chapter 2 (started). Equivalence relation (see Appendix D), congruence relation. Blackboard proofs of Theorem 2.1 and 2.2.(i).
Tue 10-12 U28 Exercises: Sheet 1.
46 (15.11.) Mon 8-10 U28 & Wed 10-12 U37 Chapter 2 (finished). Make sure that you understood (a) in which sense the definitions of class addition and class multiplication could be messy and why they are not, and (b) what is behind the equation 1+1=0 in the set of congruence classes mod 2. Try to find your own proofs for some of the laws in Theorem 2.7. The proof of Theorem 2.11 is left to you as Exercise 2.7. Chapter 3 (started). Everything up to 3.5. Try to find your own proofs for parts of 3.5. Recall what kind of object the ring of all n-times-n-matrices over a ring is and find zero divisors for n>1.
Tue 10-12 U28 & Fri 13-14 U28 Exercises: Sheet 2.
47 (22.11.) Mon 8-10 U28 & Wed 10-12 U37 Chapter 3 (finished). We considered some basic properties of and facts on functions (surjective, injective, bijective) and sets (finite, infinite), as in Appendix B of Hungerford's book. Chapter 4 (started): Everything up to Theorem 4.4 (Division Algorithm). Theorem 4.1. is somehow obsolete, because I gave a rigid introduction to polynomials similar as in Appendix G of Hungerfords book, see my slides. In particular, polynomials are denoted by f rather than by f(x). Exercise 4.2 shows that this is all consistent with the ``sum notion'' of polynomials.
Tue 10-12 U28 & Fri 13-14 U28 Exercises: Sheet 3. You are asked to write down solutions for three parts of Sheet 3 at home and to hand in your writings until Wednesday next week, 12.00. It is mandatory for the permission to the exam that at least two of these are essentially correct. You may also submit by mail it to Sven Simonsen, Campusvej 55, DK-5230 Odense M, if it is difficult to personally attend (please take care that it is here by next Wednesday). My prefered language is english, please give it a try. If this is too difficult for you, you could write in danish (and see if it is too difficult for me).
48 (29.11.) Mon 8-10 U28 & Wed 10-12 U37 Chapter 4 (finished). Most of the proofs of earlier chapters carry over to the situation of polynomial rings over a field. Sven finished the Chapter on Wednesday, including the sections on reducibility of polynomials over the rationals and the complex numbers.
Tue 10-12 U28 & Fri 13-14 U28 Exercises: Sheet 4.
49 (6.12.) Mon 8-10 U28 & Wed 10-12 U37 Chapter 5. Most of the proofs of earlier chapters carry over. Chapter 6 (started). Sven accomplished everything up to the definition of prime ideal. Most of the arguments have been generalized from earlier chapters.
Mon 14-15 U28 & Tue 10-12 U28 Exercises: Sheet 5.
50 (13.12.) Mon 8-10 U28 & Wed 10-12 U37 Chapter 6 (finished). Now we know exactly when our quotient rings are integral domains or fields, respectively. Try to set up a complete list of methods how to obtain new rings from known ones. For each method, recall how they affect ring properties such as being an integral domain or a field etc. Chapter 9 (started). We would like to have integral domains with good divisibility properties. This leads first to the definition of Euclidean domains (EDs). More general classes of rings come up: principal ideal domains (PIDs), unique factorization domains (UFDs), and rings satisfying the accending chain condition on principal ideals (ACC-on-PI). Convince yourself that they form a hierarchy (ED implies PID implies UFD implies ACC-on-PI). Study the proofs of Theorem 9.8 (ED implies PID), Lemma 9.10 (PID implies ACC-onPI), Lemma 9.11 (irreducible elements generate prime ideals in PIDs) and Theorem 9.12 (PID implies UFD): They are archetypes of basic algebra proofs.
Mon 14-15 U28 & Tue 10-12 U28 Exercises: Sheet 6. (I will not ask for mandatory exercises except those on Sheet 3.)
51 (20.12.) Mon 8-10 U28 & Wed 10-12 U37 Chapter 9 (finished).We treated the material from Section 9.1 and Section 9.2 (i.e. everything up to Theorem 9.18). Chapter 10. The main difference to the book is that we consider linear independence as an attribut of an arbitrary family of vectors. (The book is more special here and not always consistent.) Note that this depends heavily on the ground field. Recall why we do not call the compability laws (v) to (ix) in the definition of a vector space distributive laws or associative laws, etc. Observe that we proved Lemma 10.1 for an infinite sequence of vectors. Does the book version, with finite sequences, follow from this, or is it maybe better to rewrite the proof for this case? We started to apply the theory of vector spaces to the special case of extension fields and finished Section 10.1.
Mon 14-15 U28 & Tue 10-12 U28 Exercises: Sheet 7.
Wed 16-18 U144 Exercises

*: The date (dd.mm.) indicates monday of the respective week.

## Contents

This is an introductory course to abstract algebra, covering some basic facts and methods from elementary number theory and ring theory: The integers, divisibility, congruence and modular arithmetic, rings, polynomial rings, congruence in polynomial rings, ideals and quotient rings, integral domains and factorization, fields and field extensions (if time permits).

## Prerequisites

Basic knowledge on linear algebra.

## Literature

The course covers the material of several sections of the book

Abstract Algebra: An Introduction by T. W. Hungerford, 2nd edition, Saunders College Publishing (1997).

The plan is to deal with most sections of chapters 1 to 6, and, if time permits, of chapters 9 and 10. There is a follow-up course, Groups and Vectorspaces [Grupper og Vektorrum], in the 4th quarter (3rd year), relying on the same book.

Further books are in the MM510 slot in IMADA's library.

9.2.2011