Galois Theory: A Study of Cyclotomic Field ... - Scripps College

Subgroups and Fixed Fields 23
Then this means
ζ + σ 3 ζ + σ 6 ζ + σ 9 ζ + σ 12 ζ + σ 15 ζ = ζ + ζ 23 + ζ 26 + ζ 29 + ζ 212 + ζ 215
= ζ + ζ 8 + ζ 7 + ζ 18 + ζ 11 + ζ 12
is fixed by all of H 6 and Q(ζ + ζ 8 + ζ 7 + ζ 18 + ζ 11 + ζ 12 ) is the fixed field
of H 6 .
Finally, we consider the corresponding field for C 9 , namely H 9 . The
elements of H 9 = {id, σ 2 , σ 4 , σ 6 , σ 8 , σ 10 , σ 12 , σ 14 , σ 16 } fix all elements in
ζ + σ 2 ζ + σ 4 ζ + σ 6 ζ + σ 8 ζ + σ 10 ζ + σ 12 ζ + σ 14 ζ + σ 16 ζ.
This means
ζ + σ 2 ζ + σ 4 ζ+σ 6 ζ + σ 8 ζ + σ 10 ζ + σ 12 ζ + σ 14 ζ + σ 16 ζ
= ζ + ζ 22 + ζ 24 + ζ 26 + ζ 28 + ζ 210 + ζ 212 + ζ 214 + ζ 216
= ζ + ζ 4 + ζ 16 + ζ 7 + ζ 9 + ζ 17 + ζ 11 + ζ 6 + ζ 5
is fixed by all of H 9 . Again, no other σ in G fixes
ζ + ζ 4 + ζ 16 + ζ 7 + ζ 9 + ζ 17 + ζ 11 + ζ 6 + ζ 5 ,
so Q(ζ + ζ 4 + ζ 16 + ζ 7 + ζ 9 + ζ 17 + ζ 11 + ζ 6 + ζ 5 ) is the fixed field of H 9 .
This can be more easily understood in the form of a field extension lattice.
See Figure 4.2
In the lattice representation of the field extension, we can clearly see the