Exercise 8.3.3 (Subgoups of $D_4$)

Proof. Here $H\simeq C_2\times C_2$ is a subgroup of $G=\mathrm{Gal}(\sqrt[4]{2},i)\simeq D_4$.

The elements $a=(1,0),b=(0,1)$ are distinct elements of $C_2\times C_2$ of order 2 (and also $(1,1)$), so $H$ contains two distinct elements $u,v$ of order $2$, such that $\mathit{uv}$ has order $2$. The corresponding isometries $f,g$ of the square are of order $2$ in $D_4$, thus $f$ and $g$ are a rotation by $\pi$ or a reflection. If $f$ or $g$ is a rotation by $\pi$, then $u={\rho }^2$ or $v={\rho }^2$, so ${\rho }^2\in H$. In the other case, $f$ and $g$ are reflections, thus $f\circ g$ is a rotation, which means that $\mathit{uv}\in ⟨\rho ⟩$, with order $2$, thus $\mathit{uv}={\rho }^2$. In both cases, ${\rho }^2\in H$. □