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Are there any analogues of Galois Theory for complex numbers (with non-zero imaginary part)? This is motivated by Schwarz Principle. It says that if $f$ is analytic, $f$ is defined in the upper-half disk, and $f$ extends to a continuous function on the real axis, then $f$ can be extended to an analytic function on the whole disk by the formula $f(\bar{z}) = \overline{f(z)}$.

This reminds me of the Isomorphism Extension Theorem.

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Perhaps you should explain what exactly this analogue is supposed to deal with and what questions you are hoping to answer with such an analogous theory? Otherwise this question is a fishing expedition and I can think of at least three different answers, answering completely different questions. –  Alex B. Dec 23 '10 at 1:32
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In the reflection principle, you need that not only $f$ extends continuously to the real axis, but also it only takes real values on the real axis. –  Willie Wong Dec 23 '10 at 2:22
    
Perhaps this is more along the lines of a Galois connection en.wikipedia.org/wiki/Galois_connection ? But I don't really see it as analogues of Galois theory, but rather of extension theorems of all kinds; there's some in Group Theory (inducing representations, extending homomorphisms), linear algebra has isomorphism extensions, etc. –  Arturo Magidin Dec 23 '10 at 3:38
    
@Arturo: where do you see a Galois connection here? I don't see one myself. –  Pete L. Clark Dec 23 '10 at 4:52
    
@Pete L. Clark: To be honest, I'm grasping at straws because I also don't see a link to Galois Theory (I don't see a link to Galois theory from the Extension Theorem either). But perhaps along the lines of thinking of the operations of extension of restriction (to the real axis) and extension (to the plane) to get a closure operator on functions. It's weak, I admit; but to me the link to the Extension Theorem or to Galois theory seems at least as weak... –  Arturo Magidin Dec 23 '10 at 5:08
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Galois theory is useful when you have some algebraic object, and a list of tools you are allowed to use within that object. The purpose of Galois theory is to explain how far one can go only using those tools.

For example, it is impossible to create, using only the tools of +, -, *, / and nth roots, a formula for the zeroes of a general 5th degree polynomial. This is the most basic application of Galois theory. This is the Abel-Ruffini theorem:

http://en.wikipedia.org/wiki/Abel%E2%80%93Ruffini_theorem

Another application is Differential Galois theory, which tells you which integrals can be expressed in terms of elementary functions. But once again, it comes down to having only a few basic tools with which to construct an anti-derivative for a function using only elementary functions, and that is why Galois theory is useful in this application.

So, its hard to understand what your question is asking about, because you haven't really phrased your question as a Galois Theory question.

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