189-235A: Algebra 1

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Week 1 (Sept 5 and 7). This week I gave a brief motivational overview of Abstract Algebra, discussing various number systems culminating with complex numbers. Historically the latter were motivated by the algebraic solution of the cubic equation discovered by the Italian mathematicians Niccolo Tartaglia, Scipione del Ferro and Girolamo Cardano. For a delightful account of the dramatic story surrounding this discovery, see Chapter 6 of

Journey through genius: the great theorems of mathematics by William Dunham.

Cardano's solution to the cubic was a turning point because it went beyond what the ancients had been able to achieve, suggesting that there might be a lot more to mathematics than was contained in Archimedes and Euclid. It also created a compelling case for the introduction and use of complex numbers. Attempts by mathematicians and philosophers to come to terms with the ``imaginary quantities" in Cardano's formula for the (very real, both in a mathematical and ontological sense) solutions of the cubic were an important impetus for the birth of modern abstract algebra.

Week 2 (Sept 10 -14). This week I covered some basic "mathematical vocabulary", concerning proof, sets, functions, complex numbers... This is the material of Part I of the on-line notes. For some of you this was perhaps a leisurely review; on the other side of the spectrum, if you had never been exposed to this material before, I probably moved at a rather brisk pace; in that case you are advised to read Part 1 in advance of the lectures. You do not have to master the material in all its details, but even a cursory acquaintance with it will make it easier to follow the lectures.

Organisational remark. My colleagues in the mathematics department know that some of you have not yet been able to register for the course. With high probability, the problem will sort itself out by the drop-add period if, as is often the case, sufficiently many students decide to drop 235A in favor of other classes that they might be trying out at the same time. If the course is still overbooked by then, you should go see Angela White on the 10th floor of Burnside. I have been assured that everything will be done so that no one who needs the course for her or his program is turned away because of limits on registration.

Week 3 (Sept 17-21). This week we made the first steps in describing the arithmetic in Z, following loosely sections 8 and 9 of the notes. But our treatment was a bit different from the one given in the on-line notes. For instance on Monday, we introduced the set of positive integers as a set equipped with a zero element and a successor function, satisfying the axiom of induction. We then gave an inductive definition of addition and multiplication, and proved that the resulting operations satisfy the familiar rules (associativity, commutatitvity, distributivity of multiplication over addition...) The resulting (slightly pedantic) proofs are a good illustration of the general strategy of proof by induction: for more on that, see also Section 2.3. of the on-line notes. On Wednesday and Friday we turned to the Euclidean division algorithm and the algorithm for the gcd, following closely sections 8 and 9 of the notes.

Week 4 (Sept 24-28). This week I was absent and Miljan Brakocevic filled in for me. The material covered this week were the proof of the fundamental theorem of arithmetic (Section 10 of the notes) and the first steps in our discussion of congruences (Section 12 and part of Section 13.) Note that your first assignment is also due on the Monday of this week. You can either hand it in class of drop it in the assignment slot on the 10th floor of Burnside Hall. Please remember to clearly indicate your name and the name/number of the course for which you are returning this assignment.

Week 5 (Oct 1-5). Assignment 1 was handed back on Monday. In response to comments from your grader (Olivier Martin) I added some extra guidelines on the assignments page. Make sure to read them carefully and follow them in the future!

This week we discussed various topics related to congruences: notably the fact that Z/nZ is a ring, and that it is even a field if and only if n is prime. We then used this to prove Wilson's theorem and Fermat's little theorem, and then followed up with a general discussion of linear congruence equations and of the Cinese remainder theorem.

Week 6 (Oct 10-12). In this (short, because of the thanksgiving break) week, we gave a detailed discussion of the Chinese remainder theorem on Wednesday and described various approaches to primality testing on Friday, amplifying on some of the themes touched upon in Assignment 2 which was due on Wednesday.

Week 7 (Oct 15-19). The week's lectures were devoted to the theory of rings of polynomials with coefficients in a field, following more or less the development in Chapter 4 of Eyal Goren's notes.

Week 8 (Oct 22-26). Monday's course was a review session for the Midterm on Wednesday.

On Wednesday we had the midterm exam. To give you some idea, here is a sample midterm which I strongly encouraged you to work out as part of your studying for the final. Some of the questions in this practice midterm are really meant to make you think, so if you do not succeed in solving it in one hour, don't panic, this is not abnormal. Working through the practice final is excellent preparation anyway (in the same way that you might run several hours a day to prepare for a 1 mile race...)

On Friday I handed back the midterm. Many of you did quite well, with around a quarter fo the class getting in the 90-100 range. The class median was around 75, and those with garades in that zone or lower should interpret the result as a ign that they need to work harder, or organise themselves differentily, so as to get more out of the lectures and assignments and improve their performance on the final exam.

On Friday we also dicussed poslynomials and their factorisations, stating (without proof) the fundamental theorem of algebra concerning complex polynomials, and provinf that irreducible polynomials with real coefficients are either of degree 1 or 2.

Week 9 (Oct 29-Nov 2). On Monday we covered Eisenstein's criterion for irreducibility of polynomials with rational coefficients, and started talkng about congruences in polynomial rings.
On Wednesday, we focused on the case of polynomials wit coefficients in a finite field, and pushed further some of the ideas already covered in assignment 3, explaining how one might go about factoring polynomials over such fields.
Having seen the striking list of formal analogies between our study of the integers and the set of polynomials with coefficients in a field, we are now ready and motivated to embark on our next topic: general abstrract rings, of which the ring of integers, and the ring F[x] of polynomials, are two prototypical examples.

Week 10 (Nov 5-Nov 9).
We continued the discussion of abstract rings that was started in Nicolas's lecture of Friday. We discussed ideals in an abstract ring, proving the theorem that every ideal in the ring Z of integers or in the ring F[x] of polynomials over a field is always principal. We then saw examples of ideals that are not principal, for instance in the ring Z[x] of polynomials with integer coefficients, or in the ring F[x,y] of polynomials in two variables over a field F. We then discussed homomorphisms between rings, the most interesting example perhaps being the so-called Frobenius homomorphism defined on a ring R in which a prime number p is equal to 0 (such as the ring Z/pZ, the ring Z/pZ[x] of polynomials with coefficients in Z/pZ, or any quotient of the latter ring...) which sends an element a of the ring R to a^p.

Week 11 (Nov 12-Nov 17).
This week we concluded our discussion of ring theory by defining the notion of a quotient of a ring by an ideal, and describing the fundamental isomorphism theorem for rings. The notion of a quotient is one of the most subtle, important and powerful ideas in ring theory. We described one of its fundamental applications, to the construction of the finite fields with pⁿ elements, where p is a prime number and n is an arbitrary exponent (greater or equal to 1).