algebraically determining if a number is irrational or not

Question

Is it possible to use an algebraic formula, equation, concept, or principle to determine with perfect accuracy (or high precision, if not perfect) whether or not a number is rational?

An example number I have in mind is $\sqrt{937}$.

Answer 1

I think you need to consult an expert on this, but I would boldly suggest the answer is unknown. The current method is to assume the number coming from a certain algebraic equation of degree $n$, then check the decimal part of the number and see if it become linearly dependent after taking the power a few times, etc. I think for a lot of constants like $\zeta(2n+1)$ or Euler's constant, we do not actually know whether they are rational or irrational, largely because the above method fails, which in fact only tells us the weaker result that the number should be algebraic by bounding its "height". For quadratics it is easy, but if the number is not a period or coming from a non-obvious limiting process, then the answer can be tricky and difficult.

Again, excuse me if I made any mistakes. A real expert in algebraic number theory would be better to answer your question.

Answer 2

Whether a number is rational or irrational is not a matter of precision.
In the case of an algebraic number, you can find (by algebraic methods) the minimal polynomial this satisfies over the rationals. If that has degree $> 1$, the number is irrational.