Finding Roots of tenth degree polynomial

Question

I know that there are no explicit formulas to find roots for polynomials of degree higher than $4$.

I have to find all the roots of the polynomial $ f(z) = 1+z^2+z^4+z^6+z^8+z^{10}$

I found two roots, $\pm i$.

Can you suggest of a method/(s) to find all the roots.

Answer 1

Just to show some other method: For $t=z^2$ your equation becomes: $$ 1+t+t^2+t^3+t^4+t^5=0 $$ that can be factorized as: $$ (1+t)(1+t^2+t^4)=(1+t)(1+t^4+2t^2-t^2)=(1+t)(1+t^2+t)(1+t^2-t) $$ and solved.

Or noting that: $$ (1-t^6)=(1-t)(1+t+t^2+t^3+t^4+t^5) $$ we can say that the roots of $1+t+t^2+t^3+t^4+t^5=0$ are the roots of $1-t^6=0$ (six sixth roots of unit easy to find) different from $t=1$.

Answer 2

If you already have $+i,-i$, take the product of the root coefficients $(x-i)(x+i) = (x^2+1)$ and divide the polynomial with it.