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

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.

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,

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.

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.

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

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 p