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On Wikipedia one can find that transcendental number theory, or transcendednce theory is a branch of number theory. That confuses me a little since I thought that number theory is concerned with properties of positive integers and transcendence theory seems to be preoccupied with completely different questions.

Even more strangely, on Wikipedia the Schanuel´s conjecture, one of the big open problems in trancscendence theory is classified as a problem in analysis. So where does transcendence theory belong exactly?

Adam
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  • Wikipedia labels Schanuel's conjecture as number theory, just go to the "talk" tab. In any event, I believe transcendence theory is generally considered part of number theory, and often is most connected with analytic number theory. – Elchanan Solomon Sep 29 '13 at 16:18
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    You may be thinking more about combinatorics - 'number theory', particularly analytic number theory, goes well beyond the domain of the integers! Just look at the Riemann Zeta function, modular forms, $p$-adics, etc., all of which go well beyond 'integers' or even discrete concepts... – Steven Stadnicki Sep 29 '13 at 16:24
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    Dear Adam, You may want to read this answer to get a better idea of what number theory is about. Regards, – Matt E Sep 29 '13 at 17:18

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Since transcendental numbers $\alpha \in \mathbb{C}$ are the same as complex numbers which are not algebraic over $\mathbb{Q}$, and since algebraic numbers (algebraic number fields, etc.) are very properly the domain of modern-day number theory, it stands to reason that aspects of transcendental number theory are tightly interwoven with algebraic number theory.

(Questions which are seemingly just about ordinary rational integers are quite typically clarified by considering them as elements of larger algebraic number fields. For example, the question of when a prime is a sum of two squares is immensely clarified by working in the context of the Gaussian integers $\mathbb{Z}[i]$.)

Transcendental number theory interfaces quite a bit with analysis as well. For example, the classical proofs that $e$ and $\pi$ are transcendental involve a little number theory and a lot of evaluation of special integrals. I wouldn't have classified Schanuel's conjecture as a problem of analysis per se (without a sense that I was somewhat distorting matters), but you could definitely place this (and transcendental number theory in general) at an interface between algebraic number theory and analysis (and quite a few other fields as well, for example model theory and logic).

user43208
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    It's worth emphasizing too that $i$ itself is an 'irrational number', on legitimately equal footing with a number like $\sqrt{2}$... – Steven Stadnicki Sep 29 '13 at 16:25
  • @StevenStadnicki Yes, quite. For example (this is for anyone reading this), we know that $e^\pi$ is transcendental by way of association with $i^i$, together with the Gelfond-Schneider theorem: if $a, b$ are algebraic with $a \neq 0, 1$ and $b$ irrational, then $a^b$ is transcendental. – user43208 Sep 29 '13 at 16:30
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Transcendental number theory typically considers the following two types of problems:

  1. To prove that some (complex) numbers are algebraically independent. In this connection, most remarkable is Baker's theorem.

  2. To show that a particular (complex) number is transcendental. This is a special case of the above. An example is Roger Apery's proof that $\zeta(3)$ is irrational and Tanguy Rivoal's works in similar directions.

Most of these transcendental numbers are defined using analytical methods. By definition, "transcendental" means, "beyond the reach of algebra". So of course some analysis will be involved.

This topic is also closely related to Diophantine Approximation. So much so that, some introductions to the subject start with Dirichlet's theorem, which can be used for simple transcendence proofs.

Good introductions can be found from Michael Waldschmidt's homepage. Look at his pdfs and slides.