189-235A: Algebra 1

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Week 1 (August 31 to September 4).

This week 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.
Friday's lecture was devoted to a discussion of some basic notions like sets, functions between sets, the notion of an injective, surjective, and bijective function, and the notion of cardinality.

Week 2 (September 9 to 11).

Wednesday's lecture completed our discussion of set theory by discussing the philosophically profound notion of cardinality and presenting Cantor's celebrated diagonal argument, which proves that the real numbers, or the power set of an infinite countable set, are not themselves countable, i.e., cannot be placed in bijection with the set of positive integers.

We then turned our attention to the set of integers, commonly denoted Z. It is the prototypical example of an abstract ring. We discussed the principle of induction and how one might go about proving all the familiar algebraic proprties of addition and multiplication from this one axiom.

Week 3 (September 14 to 18).

This week we embarked on a discussion of the integers, rigorously proving the key facts about the integers that can be found in Euclid: the idea of division with remainder, the notion of the greatest common divisor (GCD) and the justly celebrated Euclidean algorithm for computing it, as well as the fundamental theorem of arithmetic and the fact that there are infinitely many prime numbers.

On Friday's lecture we introduced the notion of congruence modulo an integer n, and showed that the relation of congruence is an equivalence relation, and that it is compatible with the operations of addition and multiplication. This allowed us to conclude that the quotient set Z/nZ is endowed with a natural addition and multiplication inherited from the similar operations on Z

Week 4 (September 21 to 25).

This week we started by introducing the notion of an abstract ring, of which the integers and the set Z/nZ, endowed with their familiar operations of addition and multiplication, are a prototypical example.We then delved further into the structure of the rings Z/nZ of residue classes modulo n, proving the following far reaching structural characterisation of the prime numbers: An integer N is a prime number if and only if the ring Z/NZ is a field. It is thanks to results like this that we know that the integer $2^{136,279,841}-1$ (a number with more than 41 million decimal digits) is prime, and that $2^{4096}+1$ (a 1234-digit number) is composite, without knowing (as of this week) all of its prime factors. Other results that were shown were Fermat's Little Theorem, Wilson's theorem, and the relevance of quick primality tests and the hardness of factoring for public key cryptography.