Computing roots with De Moivre

Exponential form. $-1=e^{i\pi}$.

Fourth roots. $w_k=e^{i(\pi+2k\pi)/4}$ for $k=0,1,2,3$:
- $k=0$: $w_0=e^{i\pi/4}=\frac{\sqrt2}{2}(1+i)$.
- $k=1$: $w_1=e^{i3\pi/4}=\frac{\sqrt2}{2}(-1+i)$.
- $k=2$: $w_2=e^{i5\pi/4}=\frac{\sqrt2}{2}(-1-i)$.
- $k=3$: $w_3=e^{i7\pi/4}=\frac{\sqrt2}{2}(1-i)$.

Conclusion. The four roots are $\frac{\sqrt2}{2}(\pm1\pm i)$, with all sign combinations.