[Math] What are examples of unexpected algebraic numbers of high degree occured in some math problems


Recently I asked a question about a possible transcendence of the number $\Gamma\left(\frac{1}{5}\right)\Gamma\left(\frac{4}{15}\right)/\left(\Gamma\left(\frac{1}{3}\right)\Gamma\left(\frac{2}{15}\right)\right)$, which, to my big surprise, turned out to be an algebraic number, but not some decent algebraic number like $\left(\sqrt{5}-1\right)/2$, but an enormous one with the minimal polynomial of degree 120 and a coefficient exceeding $10^{15}$.

So, my question: are there other interesting examples of numbers occurred in some math problems that were expected likely to be transcendental, but later unexpectedly were proven to be algebraic with a huge minimal polynomial.

Best Answer

I don't know if Conway's constant is quite what you are looking for, as I'm not sure one would expect it initially to be transcendental or not. So, perhaps it's my bad intuition, but I was certainly surprised to learn that it is an algebraic number with minimal polynomial of degree 71.

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