# The smallest class of numbers closed under addition, multiplication, and exponentiation

Let $\def\A{\mathfrak A}\A$ be the smallest subset of $\Bbb C$ that contains the algebraic numbers and also all numbers of the form $$\sum \alpha_i^{\beta_i}$$ where the $\alpha_i, \beta_i$ are themselves elements of $\A$. This class $\A$ is much larger than the algebraic numbers. For example, it contains $2^{\sqrt 2}$, which is known to be transcendental, by the Gel'fond–Schneider theorem.

Does $\A$ have a name? Is anything known about $\A$? In particular, is it known whether $e$ and $\pi$ are in $\A$?

• I believe this is the EL numbers, see Chow 1999, pp. 441–442. – user122283 Apr 7 '14 at 20:37
• It is Chow, Timothy Y. (May 1999), "What is a Closed-Form Number?", American Mathematical Monthly 106 (5): 440–448. – user122283 Apr 7 '14 at 20:39
• Wikipedia says that the EL numbers are closed under logarithm; I don't think that $\mathfrak A$ is. But thank you for the reference, which I will try to obtain (and which I think I might already have read, but forgotten!) [Addendum: the Chow paper is available from Chow's website] – MJD Apr 7 '14 at 20:40
• $a,b\in \mathfrak{A}\implies\exists c:c=a^b\implies b=\log_ac,$ i.e., if $\mathfrak{A}$ is closed under exponentiation, it's closed under logarithms of base $a\in\mathfrak{A}$. – user122283 Apr 7 '14 at 20:44
• Yes, but if I understand Chow's definition (bottom of p.441) correctly, this is not what he means. He wants EL to be closed under $\exp$ and $\log$, where these are the base-$e$ versions. – MJD Apr 7 '14 at 20:48