>>237
つづき

4. Ambiguity by units
P8
Let qo generate p, the ideal lying under P in Z[ω], where P defines the Kummer (-Teichm¨uller) character.
Identify (Z/m)× with the Galois group of Q(ω) over Q, which we know acts transitively on primes over p in Z[ω].
6. Numerical examples
P13
[6.7] p = 11 and order m = 5 Since ω = ω5 satisfies ω^4 + ω^3 + . . . + ω + 1 = 0,
0 =((ω + 2) - 2)^4+((ω + 2) - 2)^3+ . . . +((ω + 2) - 2)+ 1 = (ω + 2)^4 + . . . + 11
The constant term 11 = (2^5 + 1)/(2 + 1) is the norm of qo = ω + 2, so
11 = (ω + 2)(ω^2 + 2)(ω^3 + 2)(ω^4 + 2)
The fifth power of the quintic Gauss sum is
γ(χ^-2_P )^5 = η ・ (ω + 2) (ω^2 + 2)^3(ω^3 + 2)^2(ω^4 + 2)^4
and the congruence for η is
-η (ω^2 + 2)^2(ω^3 + 2) (ω^4 + 2)^3 = (-1/((11-1)/5)!)5 mod (ω + 2)
Using ω = -2 mod ω + 2, this is
η ((-2)^2 + 2)^2((-2)^3 + 2) ((-2)^4 + 2)^3 =1/2^5 mod (ω + 2)
or
η ・ 6^2・ (5) ・ (7)^3 = -1 mod (ω + 2)
which simplifies to η ・ 3 ・ 5 ・ 2 = -1 mod (ω + 2) and then 3η = 1 mod (ω + 2), so η = 4 mod (ω + 2). Since
ω = -2 mod (ω + 2), this gives η = ω^2. Thus,
γ(χ^-2_P )^5 = ω^2・ (ω + 2) (ω^2 + 2)^3(ω^3 + 2)^2(ω^4 + 2)^4
and the quintic subfield of Q(ω5, ζ11) is generated over Q(ω5) by the fifth root of this.

つづく