It sometimes happens that $$\prod_{k=1}^n \frac{\Gamma (a_k/m)}{\Gamma (b_k/m)}$$ is algebraic for positive integers $m,n,a_k,b_k$. For example, $$\frac{\Gamma\left(\frac{1}{24}\right)\Gamma\left(\frac{11}{24}\right)}{\Gamma\left(\frac{5}{24}\right)\Gamma\left(\frac{7}{24}\right)}=\frac{\sqrt{6}}{\sqrt{3}-1}$$ and $$\frac{\Gamma\left(\frac{12}{42}\right) \Gamma\left(\frac{11}{42}\right)}{\Gamma\left(\frac{2}{42}\right)\Gamma\left(\frac{21}{42}\right)} = \frac{8 \sin\left(\frac\pi7\right) \sqrt{ \sin\left(\frac\pi{21}\right) \sin\left(\frac{4\pi}{21}\right) \sin\left(\frac{5\pi}{21}\right)}}{\sqrt[42]2 \; \sqrt[3]7 \; \sqrt[28]{19683}};$$ note that $\sin (\pi r)$ is algebraic for $r\in\mathbb{Q}$.
Now suppose I want to write an algorithm which takes an expression of the form $$\prod_{k=1}^n \frac{\Gamma (a_k/m)}{\Gamma (b_k/m)}$$ (one that is known to be algebraic) as input. Then
It finds that algebraic number by expressing it by rooty numbers (like $\sqrt{3+\sqrt{2}}$ etc.) and trigonometric numbers ($\sin (\pi r)$ where $r$ is rational).
The trigonometric numbers (possibly coming from Euler's reflection formula) can be further converted to rooty numbers, but we won't bother doing that for this algorithm.
The user pisco has already answered a similar question (Simplify $\Gamma\left(\frac27\right) \Gamma\left(\frac{11}{42}\right)/\;\Gamma\left(\frac1{21}\right)$ to elementary terms). However, it is unclear how can his answer be turned into the above algorithm (maybe it's not even supposed to). Meanwhile, dan fulea's answer relies on knowing the answer beforehand.
Is the algorithm possible? If so, how?
Idea
Maybe using the algdep command from PARI/GP to guess the minimal polynomial and then keep checking symbolically (as indicated in dan fulea's answer) would work – this process should terminate, but I'm not sure.
