$\mathbb{C}$ is algebraically closed, so $X^2 - i \in \mathbb{C}[X]$ has two roots counting multiplicities. Geometrically, $e^{i\theta_1}$ and $e^{i \theta_2}$ must be roots if we set $\theta_1 = \pi/4$ and $\theta_2 = 5 \pi / 4$. But:
$$\begin{align*} i^{1/2} &= i^{4/8} \tag{1} \\ &= (i^4)^{1/8} \tag{2} \\ &= (1)^{1/8} \tag{3} \\ &= \langle e^{i \pi/4} \rangle \tag{4} \end{align*}$$ (the last line is the subgroup of $(\mathbb{C}, \times)$ generated by $e^{i \pi/4}$. Which step is wrong and why? Do complex numbers not satisfy $z^{\alpha \cdot \beta} = (z^\alpha)^\beta$?
