Does anyone know if there is work done in this direction where one extends (the field) $\mathbb{Q}$ or $\bar{\mathbb{Q}}$ with certain common transcendental numbers such as $\pi$, $e$, etc. For example, can one "get away" with such a field extension rather than the full of $\mathbb{R}$ in certain proofs? And if that is the case, are there any benefits in, for example, automatic proofs in computer algebra?
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1Th notion of a transcendence-degree might be of interest to You. – Peter Melech Mar 13 '21 at 16:06
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Thanks. Indeed, I am familiar with trascendence-degree. – Jap88 Mar 13 '21 at 16:36
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1Maybe useful is What Is a closed-form number? by Timothy Y. Chow (1999; JSTOR and 8 May 1998 arXiv version), in particular his discussion of the field $\mathbb E$ of complex numbers that are closed under the operations of base-$e$-logarithm and base-$e$-exponentiation. For the type of application you're thinking of, it seems reasonable to me that the author of such a paper might cite Chow's paper (which is rather well known). – Dave L. Renfro Mar 13 '21 at 17:56
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Thanks @DaveL.Renfro. – Jap88 Mar 19 '21 at 01:30
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Here is another more recent paper that I found (when searching for Chow references) interesting: https://carma.newcastle.edu.au/resources/jon/closed-form.pdf Since this is an active research topic, I don't expect a clear answer here. – Jap88 Mar 19 '21 at 03:01
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1Here's the published version: Jonathan Michael Borwein (1951-2016) and Richard Eugene Crandall (1947-2012), Closed forms: what they are and why we care, Notices of the American Mathematical Society 60 #1 (January 2013), 50-65. See also What does closed form solution usually mean? and these questions linked to it. And maybe my answer to Expanded concept of elementary function? – Dave L. Renfro Mar 19 '21 at 15:56