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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?

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    $\begingroup$ It's crucial in number theory .... $\endgroup$ Jul 10, 2017 at 6:44
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    $\begingroup$ @Lord Shark the Unknown Analytic Number Theory... $\endgroup$
    – gone
    Jul 10, 2017 at 6:48
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    $\begingroup$ Consider $(-1)^x$ where $x>0$ is a real number. $\endgroup$
    – Wuestenfux
    Jul 10, 2017 at 6:50
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    $\begingroup$ You fail to see connections, I fail to see boundaries. Mathematics is one science, personal limitations and the (non-scientific) conditions of mathematical research enforce specialization. That's all. The fundamental theorem of algebra: every non-constant polynomial has a complex zero. One line proof: otherwise, $1/p(z)$ would be a bounded, analytic function. $\endgroup$
    – user436658
    Jul 10, 2017 at 6:52
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    $\begingroup$ Complex analysis is black magic. The proof of FTA (mentioned in above comment) becomes trivial thanks to a certain Liouville's theorem in complex analysis. There is "Complex analysis in number theory" by A. Karatsuba to further elaborate on the usefulness of complex analysis. As you may already guess, lots to do with prime number theory and the zeta function. $\endgroup$
    – AlvinL
    Jul 10, 2017 at 7:01

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Your main interests show that you want to become a professional mathematician. In these circles everyone will take it for granted that you know the basics of complex analysis. Computing real integrals using all sorts of residue tricks does not belong to the mainstream of this field.

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