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)

Abusing notation, let r = P(i) and s = Q(j) as selected roots i, j of the two polynomials P, Q, respectively, i.e., P(r)=0, Q(s)=0.

I seek to find a third polynomial R and its root k, say t = R(k), such that

(1) t = $\sqrt{1 - r^2 - s^2}$

How can R be found, knowing t? (we assume that we'll find k knowing all the roots of R)

From previously experience I have found that the degree doubles from the highest degree polynomial in the radicand.

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 $t_k$?