Dihedral Galois action on four radical roots

r

cyclically sends

α \mapsto β \mapsto - α \mapsto - β

and

s

fixes

α

while negating

β

, 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.

 Ancestors (5)