Finding a minimal polynomial for a square root of an algebraic number?

Question

I have a given monic polynomial with an algebraic root $\alpha$. How can I find the minimal polynomial with a root of $\sqrt{1-\alpha^2}$ ?

Answer 1

If the given monic polynomial is $f$, then $p=f(\sqrt{1-x^2})f(-\sqrt{1-x^2})$ has $\sqrt{1-\alpha^2}$ as a zero, so the answer will be $p$ or an irreducible factor of $p$.