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.