If I had the following: $$x!=\Gamma(x+1)=121$$We see that $x\approx5$.
But is the exact value of $x$ algebraic?
For some non-whole number $x$ input that is algebraic, I think the output of the Gamma function is not algebraic, but I am unsure.
Because there does exist some $x$ that satisfies $\Gamma(x+1)=121$, where we see $121$, the output, as algebraic, but this does not tell me much about whether or not $x$ will be algebraic.