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Complex analysis seems to work because of the interplay between algebraic geometry over $\mathbb{C}$, and analysis and topology exploiting the fact that $\mathbb{C}/\mathbb{R}$ happens to be a quadratic extension. Is there an exposition of elementary complex analysis (maybe at the level of Ahlfors's Complex Analysis) that very carefully separates these things out, perhaps pointing out at each step what would be the same or different when working over other fields or field extensions?

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    $\begingroup$ I doubt that there is no such exposition, since analysis over fields other than $\mathbb R$ or $\mathbb C$ is a specialized topic. ... I also don't think that the field structure of $\mathbb C$ is crucial for complex analysis. From my point of view, complex analysis works because $\mathbb R^2$ has 1st order elliptic operators. Ellipticity gives high degree of regularity to the solutions, and the operator being 1st order gives the ring structure to the solution space. $\endgroup$ – ˈjuː.zɚ79365 Jun 2 '13 at 7:28
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    $\begingroup$ There's too much complex analysis that generalizes to positive characteristic algebraic geometry for me to believe that statement entirely. That said, I'm not really familiar with the perspective of elliptic operators or PDE. Is there a good exposition that derives the major results in complex analysis as a special case of that theory? $\endgroup$ – Daniel McLaury Jun 2 '13 at 8:52
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    $\begingroup$ @ˈjuː.zɚ79365 It would help me too if you could suggest a good reference to understand the statement that you made. This makes me quite curious. I am quite ignorant about operator theory, but know just basic analysis at the level of Rudin(PMA), and introductory functional analysis and complex analysis. $\endgroup$ – user90041 Nov 7 '13 at 16:14
  • $\begingroup$ Complex analysis, by definition, won’t work on fields other than ℂ. There’s no need to expose parts of this unique theory separately unless you are eager for generalizations. I am sure for analytic/algebraic functions over quadratic extensions of arbitrary fields some analogue of the Cauchy–Riemann equation shows up, but it isn’t complex analysis. Do you want to generalize some other specific results, such as that any domain in ℂ in a domain of holomorphy? BTW the concept of a holomorphic function requires a metric completeness of the field, so it doesn’t exist in many examples in algebra. $\endgroup$ – Incnis Mrsi Aug 12 '14 at 9:43
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    $\begingroup$ @ChristianBlatter: I think most expositions firmly separate those things, in the sense that nobody reading such a book would be confused as to whether something was happening "because of" molecular biology or "because of" sociological factors. The situation with complex analysis texts is quite different. $\endgroup$ – Daniel McLaury Aug 11 '16 at 1:51
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A book that makes a real effort to separate the analytic and topological aspects of complex analysis is Steven Krantz and Robert Greene's Function Theory of One Complex Variable. Like most of Krantz' other textbooks,it's extremely well written and takes the student deep into a graduate course in complex analysis.The prerequisites are minimal,too-just a good careful background in calculus of one and several variables.Indeed,one of the original touches of the book is the authors emphasize the connections between multivariable calculus in the plane and complex analysis in one variable.

I think you may find it exactly what you're looking for. Also, I'd like to second the recommendation of Jones and Singerman-it's a wonderful book focusing on a modern presentation of the relationship between geometry and complex analysis. Krantz/Greene generally focuses on the purely analytic aspects of the subject-as a result, the 2 books complement each other beautifully.I think that combination will work very well for a graduate course.

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Strictly speaking the answer is "no", there is no such book. But this is perhaps a good approximation to what you are asking:

MR0890746 Jones, Gareth A.; Singerman, David Complex functions. An algebraic and geometric viewpoint. Cambridge University Press, Cambridge, 1987.

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