189-235A: Algebra 1

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Week 1 (August 27 and 29). I gave a brief motivational overview of Abstract Algebra. One of the historical origins of the subject can be traced to the algebraic solution of the cubic equation discovered by the mathematicians of the 16th Century Italian Renaissance 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.

On Friday's lecture we discussed the integers, which are the prototypical example of an abstract ring (I didn't get to the end of it, and will conclude at the start of Monday's lecture.) This is to give you a feeling for the kind of general abstract structure we are aiming to study. In the coming week, we will be focusing on the ring of integers which is in many ways, the prototypical concrete example on which the abstract notion of ring is based.

Week 2 (Sept 3 and 5). This week we made the first steps in describing the arithmetic in Z. Last Friday, 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 (or rather, indicated how one might go about proving, if one were persistent enough) 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. This week we turned to the Euclidean division algorithm and the algorithm for the gcd, which are among the most important arithmetic facts about the integers.

This development of the integers also 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 Eyal Goren's notes, which I am asking you to cover as independent reading. For some of you this might be a leisurely review; on the other side of the spectrum, if you have never been exposed to this material before, you may have to spend a bit more time reading and absorbing the material in Part 1, preferably in advance of the lectures. The sooner you are comfortable with the language, point of view, and concept of proof, the easier it will be to follow the lectures and do the assignments.

Week 3 (Sept 8-12). This week we will follow rather closely the material in section 12 in the notes. We will study the notion of congruences, which are essential to some of the most important practical application of number theory, notably to public key cryptography. This application is based on the facts that

(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 pq of two such primes p and q, it is virtually impossible to figure out what the primes p and q are, in any reasonably amount of time, even with a battery of dedicated super computers working on the problem in parallel for months (or years, or centuries).

(c) Certain operations in the rings $\mathbb Z/n\mathbb Z$ -- notably, exponentiation -- can be calculated efficiently in practice, while their inverses are extremely hard to compute in a reasonable amount of time.

We will explain (a) and (b) more thoroughly in the coming weeks, and focus a bit more on (c) this week.