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(this is basically a repost of a question I asked at M.SE last year)

Is there an explicit real algebraic number (such that we can write its minimal polynomial and a rational isolating interval) that cannot be expressed as a combination of the constants $\pi, e,$ integers and elementary functions (rational functions, powers, logarithms, direct and inverse trigonometric functions)?

In case it is an open question, can you give an example of an algebraic number such that we do not know how to express it in elementary functions?

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In your post on stackexchange you allowed $i,$ and other elemenary complex analysis: do you still want to do this? It seems if you include integration, it would be possible to express any algebraic number this way. Anyway for an algebraic number $a$ with minimum polynomial $p,$ $a = \int_{\gamma} \frac{zp'(z)}{p(z)} dz,$ where $\gamma$ is a circle with rational radius and center, containing $a$ and no other roots of $p.$ –  J. E. Pascoe Aug 19 at 4:05

2 Answers 2

I addressed this exact question in my American Mathematical Monthly paper, What is a closed-form number? Corollary 1 in that paper states that if Schanuel's conjecture holds, then the EL numbers (i.e., the numbers expressible according to your list of rules) that are algebraic are precisely the numbers expressible using radicals. So any algebraic number that is not expressible in radicals would be an answer to your question. However, Schanuel's conjecture is not known to be true. I believe that your question is still open.

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The link to your paper doesn't seem to work. –  Robert Israel Aug 20 at 6:51
    
Google still has a cached copy, and it gives "Stable URL: links.jstor.org/…; for a page 1 Jstor preview –  Philip Oakley Aug 20 at 10:58
    
The MIT webserver is a bit flaky. Probably if you try again it will work. –  Timothy Chow Aug 24 at 20:33

Try the real root $\alpha$ of $z^5 + z - 3$. This polynomial has Galois group $S_5$, so $\alpha$ is not expressible in radicals. It is expressible using some more exotic functions such as hypergeometrics, but not AFAIK using what you're calling "elementary" functions.

On the other hand, I'm pretty sure there is no proof that $\alpha$ is not, say, a rational multiple of $\pi + e$ (it is widely believed that $\pi + e$ is transcendental, but there is no proof even that $\pi + e$ is irrational).

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I think the answers to the math.SE version of the question suggest that this depends on Schanuel's conjecture. –  Qiaochu Yuan Aug 19 at 5:10
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Hermite expressed a root of any degree 5 polynomial through elliptic integrals. It would be interesting if there are roots which cannot be expressed this way. –  Misha Verbitsky Aug 19 at 6:07
    
@Misha Verbitsky: Tito Piezas III's answer to the math StackExchange question Solving 5th degree or higher equations appears to give some information relating to your question. –  Dave L Renfro Aug 19 at 14:31

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