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

Friday's lecture concluded with my beginning to write down the definition of an

This development of the integers served partly as a pretext to cover some of the basic "mathematical vocabulary", concerning proof, sets, functions, complex numbers that will be used throughout the course. This is the material of Part I of the on-line notes, which I am asking you to cover as

On Friday, Antonio Lei will be filling in for me.

Please remember to clearly indicate your name and the name/number of the course for which you are returning the assignment.

This week we will follow rather closely the material in sections 8-10 in the notes, culminating in the proof of the fundamental theorem of arithmetic (Section 10 of the notes). This theorem, which asserts that every integers can be expressed

(a) it is not so hard to produce large primes, of 100 digits or so, on a computer (or a cell phone, for that matter), and

(b) given a product

We will explain (a) and (b) more thoroughly in the coming weeks.

Time permitting, on Friday we will also embark on the first steps in our discussion of congruences (Section 12 and part of Section 13.)

On Friday of last week, Antonio introduced you to the notion of congruences, which turns out to be a very powerful piece of notation, and will furnish us with our first interesting example, beyond the prototypical one of the integers, of a ring. This week will be devoted to 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 use this to prove Fermat's little theorem, following up with a general discussion of linear congruence equations. Applications of congruences to the RSA cryptosystem will be discussed, as well as some discussion of the problem of

Have you completely absorbed the language of set theory? You probably have if you laugh at the following joke.

This week, we will begin by wrapping up our discussion of congruences by discussing a few further topics, such as Wilson's theorem, and (more importantly) the Chinese remainder theorem and its applications to efficient parallel computing.

We will then move on 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.

The week's lectures are devoted to rings of polynomials with coefficients in a field F, following more or less the development in Part 4 of Eyal Goren's notes. Highlights include the proof of the

A theme which emerges from the unique factorisation theorem (and from its proof, which is

Time permitting, we will also discuss the factorisation of polynomials over finite fields, a topic that will also be taken up partly in assignment 3.

This week was a short one, with no lecture on Monday because of the Thanksgiving holiday, and the midterm exam on Wednesday. In Friday's lecture I gave corrections for the midterm exam, and we had time in the second half to prove a new, and important, theorem about the field

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 move to the next level of abstraction and discuss general rings, of which the integers, and the ring F[x] of polynomials, are two prototypical examples.

This week was largely devoted to covering Part 5 of Eyal Goren's notes. After briefly recalling the general ring axioms (which have already been seen before, of course) and some of their basic consequences, we introduce the basic notion of an

We proved the theorem that every ideal in the ring

Then, on Friday, I discussed the notion of products of ideals in rings like Z[sqrt(-5)], and explained how the non-unique factorisation of the number 6 into irreducible elements turned into an ostensibly unique factorisation of the principal ideal generated by 6, into four ideals. Very little was proved in this lecture, but this calculation gives a glimpse of one of the historical motivations of the introduction of ideals, by Dedekind. The terminology "ideal" mirrors Dedekind's notion of an "ideal number"---these are extra elements that one has to add to the set of irreducible elements of the ring Z[sqrt(-5)] under multiplication, in order to salvage a statement like unique factorisation (which now becomes unique factorisation into ideals, and not irreducible elemments.)

On Monday we introduced the notion of the quotient of a ring by an ideal and worked out a few examples. The notion of quotient (of one algebraic structure by another) represents a pinnacle of abstraction, as far as what you will see in this course. The best way to become familiar and comfortable with the notion is to work out many examples. In fact, you have already worked with quotients before in this class (a bit like Monsieur Jourdain in Molière's "Le Bourgeois Gentilhomme", who discovered he had been speaking in prose without ever being aware of it) when you performed calculations with the congruence rings Z/nZ, and the rings F[x]/(f(x)).

On Wednesday we discussed homomorphisms of rings, and proved that the

This week we embarked on a new topic, namely, the theory of groups. These new mathematical structures are defined by a much simpler set of axioms than rings: a group is just a set endowed with a

The most important examples of groups which we discussed were

• the groups

• the permutation group

• the dihedral group

We then described the notion of a homomorphism from one group to another, and discussed the important case of a homomorphism from a finite group

Further examples of homomorphisms were given, for instance, various examples of homomorphisms from

This week we discussed the notion of

We then discussed the structure of the set

This led to a study of the all-important nottion of the conjugacy relation in a group, and to the notion of

On Monday we continued briefly with our discussion of conjugacy classes, normal subgroups, and their associated quotients, by describing a few illustrative examples, namely,

• the dihedral group of order

• the symmetric group

We stated and proved the first isomorphism theorem for groups, and explained how it could be use to identify the structure of

On Wednesday and Friday, I was out of town, and Antonio Lei filled in for me. His lectures followed sections 30-31 of the online notes, and were devoted to the all-important notion of an

This week will be devoted to covering a few topics in group theory that are left in the notes, including sections 32, 33.4, and 33.5.

This week will consist of two consecutive Mondays from the point of view of the McGill calendar, and hence we will have two hours of lecture, but on Monday and Tuesday rather than on Monday and Wednesday!

These two lectures will be devoted to review of the material. I strongly advise you to work out a lot of problems from the posted "practice finals" in the "assignments" web site, and to come to these two lectures armed with precise questions! You will get a lot more out of listening to solutions to problems you have always spent time struggling with yourself (even if with only partial success.)

In addition, there will be extra review sessions by the TA's that week, with the following schedule:

Monday, December 2, 3:30-5:00. Dylan Atwell-Duval, in BH 1018.

Tuesday, December 3, 12:30-2:30. Henri Darmon, BH1111 or 10th floor lounge of Burnside Hall.

Tuesday, December 3, 4:00-6:00. Bahare Mirza, in Burnside Hall lounge (10th floor.)

Wednesday, December 4, 3:30-5:30. Henri Darmon, in Burnside Hall lounge (10th floor).