TITLE: Some Non-Trivial Algebraic(Rational) Number
QUESTION [7 upvotes]: 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)?

REPLY [9 votes]: Let $E/\mathbb{Q}$ be an elliptic curve, let $\Omega$ be its real period (which is the value of an elliptic integral that's easy to write down using the equation of $E$), and let $L(E,s)$ be the $L$-series of $E$, which we know by Wiles' theorem has an analytic continuation to all of $\mathbb{C}$. Then $L(E,1)/\Omega$ is a rational number, but I'd say that that's not so easy to prove. Even harder, suppose that $E(\mathbb{Q})$ has rank 1, let $P$ be a non-torsion point, and let $\hat h(P)$ be the canonical height of $P$. Then
$$
  \frac{L'(E,1)}{\Omega\cdot\hat h(P)}
$$
is a rational number. In this latter problem, we don't actually know that $\hat h(P)$ is not itself rational (this is unknown for even a single example), but the consensus seems to be that $\hat h(P)$ should be transcendental if it's not zero.