Dihedral Galois action on four radical roots
= Dihedral Galois action on four radical roots
If $r$ cyclically sends $\alpha\mapsto\beta\mapsto-\alpha\mapsto-\beta$ and $s$ fixes $\alpha$ while negating $\beta$, then $r^4=s^2=1$ and $srs=r^{-1}$. These automorphisms realize $D_8$ as the Galois group of the corresponding quartic splitting field.