The set of constructible numbers

https://en.wikipedia.org/wiki/Constructible_number

is the smallest field extension of $\mathbb{Q}$ that is closed under square root and complex conjugation. I am looking for an algorithm that decides if two constructible numbers are equal (or, what is the same, if a constructible number is zero).

As equality of rational numbers is trivial, this algorithm probably needs to reduce the complexity of the involved number in several steps.

Does such an algorithm exist? If yes, what does it look like?

A Course in Computational Algebraic Number Theory(Springer 1993 / GTM 138), specifically §4.2. But this is merely one possible reference among many: the subject is really quite standard, and googling "algorithm exact algebraic arithmetic" returns various relevant results (which probably themselves point to other references). More abstractly, Rabin's theorem on the algebraic closure can also be relevant. $\endgroup$ – Gro-Tsen Mar 16 '17 at 21:08