TITLE: Determine if a number is algebraic.
QUESTION [6 upvotes]: I recall there is an algorithm involving lattices that would tell me if a given number(approximation) is an algebraic number or not, along with the exact algebraic form of the number. Is there any tool (Gap, Mathematica, python package, etc) that has an implementation of this?
I have numbers which I can approximate to any degree of accuracy I want and I would like to know if they are algebraic or not.

REPLY [4 votes]: PARI has a function algdep which does exactly what you describe:

https://pari.math.u-bordeaux.fr/dochtml/html/Vectors__matrices__linear_algebra_and_sets.html#algdep

SageMath wraps that PARI function, also under the name algdep:

https://doc.sagemath.org/html/en/reference/rings_standard/sage/arith/misc.html#sage.arith.misc.algdep

Sort of related, Ordner is a directory of real numbers:

http://fungrim.org/ordner/

Ordner was introduced in a blog post by Fredrik Johansson:

http://fredrikj.net/blog/2019/09/what-is-the-most-common-real-number/