Every problem about algebraic-ness (rational-ness) of numbers that I have seen is in one of the below types:

- The number is algebraic(rational) and proving that it is algebraic(rational) is trivial, like proving that $\sqrt{7+\sqrt{2}}$ is algebraic.
- The number is not algebraic, and $\cdots$.

My Question:

- Is there any example of an algebraic(rational) number which is not trivial to prove that the number is algebraic(rational)?

Conway's Constantmight be an example: It is algebraic, the root of a polynomial of degee $71$. $\endgroup$ – Joseph O'Rourke Apr 5 '15 at 17:13thoseare trivial? $\endgroup$ – KConrad Apr 6 '15 at 15:01