$\DeclareMathOperator\GF{GF}$Consider the following expression:

$$ \sqrt{a_1} \pm \sqrt{a_2} \pm \dots \pm \sqrt{a_n} = 0, $$

where $a_1, \dots, a_n$ are positive integers. We want to find the number of ways to change each $\pm$ into $+$ or $-$, so that the equation is true.

One way to approach this problem is to pick a sufficiently large random prime number $p$, then find all the square roots in $\GF(p^2)$ (they always exist), and then count the number of ways to satisfy the equation in $\GF(p^2)$.

With high probability, the answer in $\GF(p^2)$ will be the same as in $\mathbb R$. But at the same time, it seems much harder to actually recover such combinations in $\mathbb R$ if they're known in $\GF(p^2)$.

That being said, assume that you know for sure that a particular combination of square roots in $\GF(p^2)$ satisfies the equation. Is there any way to, using this information, recover any such combination that satisfies it in $\mathbb R$?

I guess, what I'm actually asking is, given $p$, I want to algorithmically map each number from $1$ to $p-1$ to one of its two square roots in $\GF(p^2)$ so that by substituting each $\sqrt{a_i}$ via this mapping in the expression above, I will be able to check with high probability that this specific way of changing $\pm$ to $+$ or $-$ is also valid in $\mathbb R$.