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 concluded with my beginning to write down the definition 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.