248 Circle Division
event. By Theorem 21.9, the Galois group Γ of Q(ζ ) : Q has order 4 and comprises
the Q-automorphisms generated by the maps
ρ
k
: ζ 7→ ζ
k
for k = 1,2,3,4. The group Γ is isomorphic to Z
∗
5
by the map ρ
k
7→k (mod 5). There-
fore ρ
2
has order 4 in Γ, hence generates Γ, and Γ is cyclic of order 4.
The extension is normal, since it is a splitting field for an irreducible polynomial,
and we are working over C so the extension is separable. By the Galois correspon-
dence, any rational function of ζ that is fixed by ρ
2
is in fact a rational number.
Consider as a typical case the expression α
1
above. Write this as
α
1
= ζ + iρ
2
(ζ ) + i
2
ρ
2
2
(ζ ) + i
3
ρ
3
2
(ζ )
Then
ρ
2
(α
1
) = ρ
2
(ζ ) + iρ
2
2
(ζ ) + i
2
ρ
3
2
(ζ ) + i
3
ζ
since ρ
4
4
(ζ ) = ζ . Therefore
ρ
2
(α
1
) = i
−1
α
1
so
ρ