As part of working with dot products of vectors in an algebraic field, having previously solved the algebraic quotient, I need to find the algebraic square root as shown in (1)

Let $r_i$ be a root of polynomial P and $s_j$ be a root of polynomial Q i.e., P($r_i$)=0, Q($s_j$)=0.

I seek to find a third polynomial R and its root $t_k$, such that R($t_k$)=0, so that

(1) $t_k$ = $\sqrt{1 - r_i^2 - s_j^2}$

is satisfied. How can R be found, knowing $t_k$?