Find the least degree Polynomial whose one of the roots is $ \cos(12^{\circ})$ Find the least degree Polynomial with Integer Coefficients whose one of the roots is $ \cos(12^{\circ})$
My Try: we know that $$\cos(5x)=\cos^5x-10\cos^3x\sin^2x+5\cos x\sin^4x$$ Putting $x=12^{\circ}$ and Converting $\sin$ to $\cos$ we have
$$\frac{1}{2}=x^5-10x^3(1-x^2)+5x(1-2x^2+x^4)$$ $\implies$
$$32x^5-40x^3+10x-1=0$$
Is this the Least degree? Please let me know.

Looking at function $$f(x)=32x^5-40x^3+10x-1$$ by inspection $x=\frac 12$ is a root. So $$f(x)=(2x-1)(16 x^4+8 x^3-16 x^2-8 x+1)$$ Now, you have a quartic which can be solved with radicals and integers.
