189-235A: Algebra 1

Blog

Week 1 (January 7 and 9). The first week was devoted to a brief motivational overview of the subject, giving a discussion of complex numbers, their cartesian and polar representations, a proof of the Euler identity $e^{i\theta}= \cos(\theta) + i \sin(\theta)$, and three different proofs of the fundamental theorem of algebra asserting that every complex polynomial has a complex root. The discussion of the latter topic is taken from Section 1.2. of the book of Romik, and a delightful account of the Euler identity can be found in this mathologer video.

From now on we will be following the book of Stein very closely. (The other texts are for the enjoyment and edification of the more hardy complex analysis enthusiasts, and while they might be occasionally be referred to in the blog they will not be strictly required for the course.)

Week 2 (Jan 14 and 16). This week's lectures were devoted to Chapter 1 in the book of Stein, giving a careful discussion of convergence of complex valued functions of a complex variable, power series, and the notion of integration along curves. We will prove a few special cases of Cauchy's important theorem that the integral of a complex valued function along a closed curve vanishes when the function is holomorphic on a region in the complex plane containing the curve and its interior.

Week 3 (Jan 21 and 23).

Week 4 (Jan 28 and 30).

Week 5 (Feb 4 and 6).