Rewritting square root of non integer as a recurrence with integral coefficients.

Question

Let's say we have a number $q \in \Bbb Q-\Bbb Z$, and we take the square root of it: $\sqrt q$.

When is it possible to write $\sqrt q$ in the form$$a_1=c_1+b_1\sqrt[k_1]{a_0}$$ $$a_n=c_n+b_n\sqrt[k_n]{a_{n-1}}$$

For integers $n,b_n,c_n,k_n,a_0$?

Answer 1

With integer coefficients, taking nested radicals preserves the property of being an algebraic integer. Square roots of non-integer rational numbers are not algebraic integers, so what you ask (if I interpret the question correctly) is not possible.

If you mean, "what are the relations between nested radicals over $\mathbb{Q}$", I don't think there is any theory to detect those other than computing minimal polynomials and testing for a common zero. If you want real radicals then further testing is needed to see if a potential solution comes from a real solution, which is likewise a computational task with relatively little theory (inequalities, simple Galois theory) to provide guidance.