I'm going to do a math PhD next year. My main interests in mathematics are composition algebra, algebraic topology, and group theory.
Now I have the opportunity to take a one-semester course in complex analysis, but I am not sure if it is relevant to my research interests.
My understanding of complex analysis is that it is mostly about Laurent series, Cauchy's integral theorem, residue theorem, and evaluating weird-looking real integrals using these techniques. But I fail to see its connection to algebra.
I am sure that it is a beautiful theory and that it is also an important set of tools for applied mathematicians and engineers, but is there anything useful I can take from complex analysis as an algebraist?