The nontrivial real-conjugation automorphism of the eighth cyclotomic field

Let $K=\mathbb{Q}(\omega )$, where $\omega =e^{i\pi \mathrm{/}4}$. The class $[5{]}_8$ is a unit modulo $8$, so the cyclotomic Galois group supplier $[\mathbb{Q}({\zeta }_n):\mathbb{Q}]=\phi (n)$ and $\mathrm{Gal}(\mathbb{Q}({\mu }_n)\mathrm{/}\mathbb{Q})\cong (\mathbb{Z}\mathrm{/}n{)}^{\times }$ gives the unique $\mathbb{Q}$-automorphism $\bullet$ with ${\omega }^{\bullet }={\omega }^5=-\omega$. It fixes $i={\omega }^2$ and sends $\sqrt{2}=\omega -{\omega }^3$ to $-\sqrt{2}$. Complex conjugation and $\bullet$ commute, as is checked on the generator $\omega$. The four embeddings of $K$ are the identity, complex conjugation, $\bullet$, and their composite, because the Galois group consists of the four maps $\omega \mapsto {\omega }^j$ for $j=1,3,5,7$. Since $\bullet$ fixes $1\mathrm{/}2$ and maps $\omega$ into $\mathbb{Z}[\omega ]$, it preserves $D=\mathbb{Z}[\omega ,1\mathrm{/}2]$.

This is the nontrivial real-conjugation embedding used in the scaled-grid conditions; it is distinct from complex conjugation.