Decomposing $z^4 + 1$, where $z \in \mathbb{C}$ I have learned that there is this equation called cyclotomic equation. It appears that
$$z_n - 1 = (z - z_0)\cdot(z-z_1) \dotsc (z-z_{n-1})$$
where $z_i$ satisfy $z^n = 1$. $(z \in \mathbb{C})$
Now I would like to expand $z^4 + 1$. Is it true that in this case, we have
$$z^4+1 = (z + z_0)(z+z_1)(z+z_2)(z+z_3)$$
where $z_{\nu} = \exp\left[{i(\frac{\pi}{4} + \frac{2\pi\nu}{4}})\right]$? Or should I change the sign to negative?

Normally we would write the factorisation with minus signs, with the plus-sign expression being invalid, but here the expressions with plus and minus signs are equivalent: $z_0=-z_2$ and $z_1=-z_3$, so swapping for minus signs only permutes the roots, leaving the polynomial product unchanged.
